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Archive of Applied Mechanics

Draft Manuscript for Review

An Enha nced Scaled Boundary Finite Element Method for Linear Elastic Fracture

Journal:For Archive Peer of Applied Mechanics Review Manuscript ID AAM-16-0601.R1

Manuscript Type: Original

Date Submitted by the Author: 09-Jul-2017

Complete List of Authors: Egger, Adrian; Eidgenossische Technische Hochschule Zurich, Civil, Environmental and Geomatic Engineering; Chatzi, Eleni; Institute of Structural Engineering, Department of Civil, Environmental and Geomatic Engineering Triantafyllou, Savvas

Scaled Boundary Finite Element Method (SBFEM), Extended Finite Element Method (XFEM), Linear Elastic Fracture Mechanics (LEFM), Stress Intensity Keywords: Factors (SIFs), Block Hamiltonian Schur Decomposition (HSchur), Super Convergent Patch Recovery Theory (SPR)

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1 Arch Appl Mech manuscript No. 2 (will be inserted by the editor) 3 4 5 6 7 8 An Enhanced Scaled Boundary Finite Element Method for 9 Linear Elastic Fracture 10 11 12 Adrian Egger Eleni N. Chatzi Savvas P. · · 13 Triantafyllou 14 15 16 17 18 Received: date / Accepted: date 19 20 For Peer Review 21 Abstract A blocked Hamiltonian Schur decomposition is herein proposed for the solution 22 process of the Scaled Boundary Finite Element Method (SBFEM), which is demonstrated to 23 comprise a robust simulation tool for Linear Elastic Fracture Mechanics (LEFM) problems. 24 By maintaining Hamiltonian symmetry increased accuracy is achieved, resulting in higher 25 rates of convergence and reduced computational toll, while the former need for adoption of 26 a stabilizing parameter and, inevitably user-supervision, is alleviated. 27 The method is further enhanced via adoption of superconvergent patch recovery theory 28 in the formulation of the stress intensity factors. It is shown that in doing so, supercon- 29 vergence, and in select cases ultraconvergence, is succeeded in the Stress Intensity Factors 30 (SIFs) calculation. Based on these ﬁndings, a novel error estimator for the stress intensity 31 factors within the context of SBFEM is proposed. 32 To investigate and assess the performance of SBFEM in the context of linear elastic 33 fracture mechanics, the method is contrasted against the Finite Element Method (FEM) and 34 the eXtended Finite Element Method (XFEM) variants. The comparison, carried out in terms 35 of computational toll and accuracy for a number of applications, reveals SBFEM as a highly 36 performant method. 37 38 Keywords Scaled Boundary Finite Element Method (SBFEM) Extended Finite Element 39 Method (XFEM) Linear Elastic Fracture Mechanics (LEFM) · Stress Intensity Factors 40 (SIFs) Block Hamiltonian· Schur Decomposition (HSchur) Super· Convergent Patch 41 Recovery· Theory (SPR) · 42 43 44 This research is supported by the Swiss National Science Foundation (SNSF) Research Grant, project # 45 200021 153379. 46 A. Egger E.N. Chatzi · 47 Chair of Structural Mechanics, Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland Tel.: +41 44 633 35 76 48 E-mail: [email protected] 49 S.P. Triantafyllou 50 Faculty of Engineering, University Park, Nottingham, NG7 2RD, UK 51 Tel.: +01 15 951 41 08 52 E-mail: [email protected] 53 54 55 56 57 58 59 60

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1 2 3 2 Adrian Egger et al. 4 5 1 Introduction 6 7 Since its inception, the finite element method (FEM) has been advanced to handle a mul- 8 titude of structural analysis problems ranging from linear to nonlinear, static to dynamic, 9 fracture and contact problems amongst others ([1], [2], [3]). Pertaining to the field of frac- 10 ture mechanics, it has been demonstrated that modeling of damage-related phenomena such 11 as crack initiation, crack propagation and delamination can successfully be accomplished by 12 means of the FEM [4]. Further, with the introduction of singular elements, stress singular- 13 ities about crack tips may be more accurately quantified [3]. Nonetheless, various undesir- 14 able characteristics persist, which render this method computationally prohibitive for more 15 involved analyses. Extensive mesh refinement and remeshing, as well as mesh-dependent 16 projection errors, result in decreased accuracy and increased implementational complexity. 17 As a result, alternative methods have been pursued such as the Boundary Element Method 18 (BEM) ([5], [6]), the Discrete Element Method (DEM) [7], meshless methods [8], cohesive 19 elements [9] and phase field models [10]. In general, these alternatives are not supported in 20 commercial structural analysisFor software packagesPeer and are moreReview likely to be encountered in 21 special purpose applications. 22 In mitigating the aforementioned issues of the FEM, the eXtended Finite Element Method 23 (XFEM) [11] was introduced. A combination of the Level Set Method (LSM) [12] for track- 24 ing discontinuities, combined with jump and tip enrichment for the displacement solution, 25 is utilized to decouple both weak (material interfaces) and strong (cracks) discontinuities from the mesh. The popularity of XFEM, currently available in numerous commercial soft- 26 ware packages, may largely be attributed to the fact that the core steps of FEM as well as 27 its robustness are left intact, which facilitates implementation and end-user understanding 28 alike. Although this method has aggregated significant academic interest and resolved most 29 shortcomings of FEM-based damage simulation, a number of issues still remain. The most 30 notable include the extension to 3D problems when using the LSM ([13],[14]), the larger 31 condition number of the stiffness matrix ([15],[16]) and the a priori assumption for the type 32 of tip enrichment [17]. Additionally, special integration schemes are required in order to 33 determine the stress intensity factors (SIFs). 34 An alternative approach, namely the Scaled Boundary Finite Element Method (SB- 35 FEM), attempts to fuse the advantageous characteristics of FEM and BEM into one method, 36 while simultaneously introducing features that facilitate the modeling of damage-related 37 phenomena. This method’s key feature, the introduction of a scaling center, has long been 38 exploited for the solution of electric field problems [18]. After refinement and application to 39 solid mechanics it was initially termed the “Cloning Algorithm” by Dasgupta [19], where 40 it was used to model wave propagation in unbounded soils and found applications in soil- 41 structure interaction and earthquake engineering. Subsequent work by Wolf resulted in a 42 similar formulation named the “infinitesimal finite element method” [20]. Wolf and Song, 43 first coined the term SBFEM, once the derivations for various types of problems were stan- 44 dardized using the minimum weighted residual method ([21],[22]). Deeks and Wolf em- 45 ployed a virtual work formulation, facilitating the practical implementation of the method 46 [23]. Though early work on SBFEM was intended for modeling unbounded domains, it soon 47 became apparent that it is at least equally effective for bounded domains [22]. 48 By introducing a scaling center inside the domain, the Cartesian reference system is 49 cast into a polar-like formulation. This can be exploited for determining an analytical solu- 50 tion along the radial direction, thereby only necessitating discretization along the boundary. 51 The trade off is the solution of a Hamiltonian eigen-problem, which essentially doubles 52 the problem size. General purpose eigen and Schur decompositions do not respect this spe- 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 3 4 5 cial structure and thus numerical issues arise. The eigen-problem for generating the stiff- 6 ness matrix in SBFEM may also be found in optimal control problems, as they both stem 7 from an algebraic Riccati equation. Until recently, the solution to this Hamiltonian eigen- 8 problem in Schur form has proven difficult to solve; developing a O(n3) backward-stable, 9 structure-preserving algorithm, as posed in [24], came to be known as Van Loan’s curse. 10 Chu et al. [25] proposed a new method, which seemed to resolve this issue, however it en- 11 countered numerical difficulties when dealing with tightly-packed clusters of eigenvalues, 12 which is generally the case in SBFEM. As a response Mehrmann et al. [26] augmented this 13 method and subsequently proposed a “new block method for computing the Hamiltonian 14 Schur form”, thus mitigating the aforementioned issues. Additionally, the blocked nature 15 of the algorithm favors computational efficiency as it can better exploit cache memory. If 16 further takes advantage of the solution symmetry, as all eigenvalues are symmetric about 17 the real and imaginary axis. The latter results in a reduction of the computational toll as 18 compared to a standard Schur decomposition. The main benefit to SBFEM however lies in 19 its robustness, when adopted for the solution process. Previously, due to machine precision, 20 a mesh dependant stabilizingFor parameter hadPeer to be added to Review the input of the eigen-problem 21 ([21],[27]). Its optimal value is not known a priori. The method developed in [26], applied to 22 SBFEM and benchmarked herein, is shown to alleviate this SBFEM related issue, provide 23 improved convergence rates and cut down on computational cost. 24 A strength of SBFEM is that damage-related phenomena can be accounted for in an 25 elegant manner. Solution of the eigen-problem leads into straightforward identification of 26 two singular eigenvalues and corresponding eigenmodes, which are then used to compute the Stress Intensity Factros (SIFs). Adoption of a Schur decomposition for solution of the 27 underlying eigen-problem ensures a better treatment of repeated eigenvalues than a stan- 28 dard eigen-decomposition ([28],[29]). Furthermore, the Schur decomposition allows to in- 29 corporate multiple types of crack tip singularities with enhanced accuracy [30], particularly 30 for cases where the crack resides on material interfaces. This in turn allows for an im- 31 proved quantification of the SIFs, which is particularly meaningful for crack propagation 32 ([31],[32],[33]). However, in order to quantify the quality of each step and the associated 33 discretization level, an error estimator for the SIFs need be defined. In this paper, an efficient 34 and elegant, a posteriori error estimator for the SIFs is presented, based on previous work 35 by Deeks [34], where stress recovery was introduced to SBFEM utilizing Super-convergent 36 Patch Recovery theory (SPR) ([35],[36]). Although adaptivity schemes for SBFEM have 37 been introduced ([37],[38]), they focus on error estimators for the whole domain and not a 38 single, local quantity of interest such as the SIFs. This proposed error estimator sets itself 39 apart in that it does not require the solution of a dual problem ([39],[40]), which necessi- 40 tates a second analysis, thus increasing its computational toll. Additionally, it is applicable 41 to not only one [41], but all cracking modes. A further benefit of the application of SPR to 42 the calculation of the SIFs in SBFEM is that they are shown to be superconvergent, and in 43 select cases ultraconvergent. 44 To date, no comprehensive, direct comparison between SBFEM and XFEM exists, though 45 in [33] a polygon-based SBFEM is contrasted to XFEM, while [42] compares XFEM with 46 SBFEM tip enrichment to standard XFEM. In this paper, numerical examples are treated, 47 which elaborate on the previous findings and fill in the remaining voids. First, the conver- 48 gence behavior of SBFEM alone is treated extensively for displacements, stresses and SIFs, 49 which in itself has not been adequately reported in existing literature. Secondly, the con- 50 vergence behavior for the SBFEM-computed SIFs is compared against those computed via 51 XFEM in two separate numerical examples, providing novel insight into which method is 52 better suited for Linear Elastic Fracture Mechanics (LEFM) applications. 53 54 55 56 57 58 59 60

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1 2 3 4 Adrian Egger et al. 4 5 Additional topics outside the scope of this paper also include the treatment of dynam- 6 ics. Solutions based on continued fractions [43], frobenius series expansion [44], as well 7 as the spectral element method [45] have been proposed. Nonlinearity in SBFEM has only 8 recently been introduced. Material nonlinearity, i.e., elastic-plastic behavior assuming asso- 9 ciative plasticity, has been demonstrated in [46], while material and geometric nonlinearity 10 has been incorporated in [47]. In [48], a nonlinear heat transfer problem was solved via the 11 homotopy analysis method (HAM). Further, SBFEM has been successfully coupled with 12 other numerical methods such as FEM [49], BEM [50] and meshless methods [51], in order 13 to exploit SBFEM’s unique abilities to reduce the problem dimension by one. In addition, 14 SBFEM has been used to enhance other numerical methods. This includes, but is not limited 15 to, an XFEM variation in which SBFEM was used for crack tip enrichment [42] as well as 16 the scaled boundary isogeometric analysis (SBIGA) [52], in which the scaled boundary con- 17 cepts are fused with non uniform rational b-splines (NURBS) from isogeometric analysis. 18 A novel take on SBFEM, which relies on concepts from spectral element methods, results 19 in decoupled coefficient matrices for each node [53]. The main benefit of this method is that 20 it bypasses the need for an eigen-decompositionFor Peer in standard Review SBFEM, and has thus the po- 21 tential to be significantly more computationally efficient and parallizable, especially when 22 considering structures with more degrees of freedom (DOF) such as 3D problems. 23 24 25 26 27 28 29 30 31 32 33 34 This paper is organized as follows: Section 2 introduces the theoretical background of 35 this paper. In 2.1 an abridged formulation of SBFEM theory is treated. The standard pro- 36 cedure for calculating SIFs is reviewed in 2.2. The novel proposed enhancements to the 37 SIF calculation procedure, as well as the accompanying error estimator, is presented in 2.3. 38 The newly proposed Hamiltonian Schur decomposition (HSchur), advocated for adoption in 39 SBFEM, is treated in 2.4. Four applications are demonstrated in section 3. In the first numer- 40 ical example (3.1), the issues associated with the current solution procedure in SBFEM are 41 discussed, while demonstrating the ability of the proposed HSchur decomposition to over- 42 come these, based on the example of an edge cracked square plate subjected to bending. The 43 second numerical example (3.2), provides an exact solution for displacements, stresses and 44 SIFs, establishing a new benchmark. The convergence behavior of SBFEM in terms of dis- 45 placements, stresses and SIFs is discussed, offering a unique insight into the method. In the 46 third numerical example (3.3), SBFEM is contrasted to XFEM and the standard FEM, based 47 on floating point operation counts. This type of extensive contrasting to XFEM in terms of 48 performance and computational complexity has not so far been offered in literature. The 49 fourth numerical example (3.4), a slant crack in a finite body, investigates the performance 50 of SBFEM in the worst case scenario and evaluates how it performs compared to XFEM and 51 the standard FE-based approach. Section 4 summarizes the findings and provides direction 52 for future research. 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 5 4 5 ¯ 6 t y

7 Γ

8 O Domain Ω 0 x

9 u¯ Boundary P(xˆ, yˆ) 10 Legend: 11 P(xˆ, yˆ) point in domain 1 e = ξ − Domain Ω P(x, y) point on boundary 12 1 = line element (u¯ , t¯) boundary conditions of domain = 13 (u¯ e, t¯e) boundary conditions of elements (i.e. discretized boundary) = 14 nodes of line element with coordinates (xn, yn) = O scaling center and origin of normalized radial coordinate ξ 15 = (ξ,η) scaled boundary coordinates with 0 ξ 1 and 1 η 1 16 1 = ≤ ≤ − ≤ ≤ DOFi degree of freedom of each node in global (x,y)-direction η = 17 (x, y) global coordinate system = 18 DOFy

19 u¯ e DOFx 20 ForP(x, y ) Peer Review 21 t¯e 22 Fig. 1 Introduction of SBFEM speciﬁc discretization of the bounded domain. 23 24 2 Theory 25 26 This paper deals with 2D elastostatics (Fig. 1), which is governed by the following strong 27 form. 28 29 equilibrium: Lσ + b = 0 in Ω (1a) 30 constitutive: σ = Dε in Ω (1b) 31 T 32 compatability: ε = L u in Ω (1c) 33 boundary conditions: u = uˆ on Γu (1d) 34 σ n = tˆ on Γt (1e) 35 · 36 L is the linear differential operator, D the constitutive matrix, σ represents stress, ε strains, 37 u displacements and b body forces.u ˆ, tˆ are the prescribed displacements and tractions re- 38 spectively and n is the normal vector. 39 Firstly, the SBFEM equations for 2D elastostatics will be derived based on a virtual 40 work approach. Next, the solution procedure in SBFEM is described. Subsequently, the 41 calculation of SIFs is presented, augmented by the application of SPR theory. Finally, the theory behind the new block method for computing the Hamiltonian Schur form is detailed. 42 43 44 2.1 Summary of SBFEM 45 46 The fundamental difference between SBFEM and other numerical methods is the intro- 47 duction of a scaling center O. In all but a few special cases, this scaling center must be 48 directly visible from any point on the domain boundary. By introducing the scaling center, 49 the Cartesian coordinate system is transformed into one similar to polar coordinates. It is in 50 this process that a radial coordinate ξ and a tangential coordinate η are introduced (Fig. 1). 51 Assuming that for SBFEM an analytical solution can be found in radial direction ξ, 52 which is later shown to exist (section 2.1), only the tangential direction η need be discretized 53 54 55 56 57 58 59 60

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1 2 3 6 Adrian Egger et al. 4 5 6 7 ξ ξ ξ 8 0 1 ξ1 ξ2 1 ∞ 9 10 11 scaling center system domain 12 boundary node ξ-axis 13 14 Fig. 2 Possible domain types using SBFEM. 15 16 in the ﬁnite element sense. Depending on the bounds for ξ unbounded domains (1 ξ ∞), 17 ≤ ≤ bounded domains (0 ξ 1) and bounded domains with similar boundaries (ξ1 ξ ξ2) 18 can be accounted for≤ (Fig.≤ 2). ≤ ≤ 19 By choosing to discretize in radial direction along the boundary, the dimension of the 20 problem is reduced by one.For Consequently, Peer the information Review contained within each element 21 on the boundary encompasses that of the triangular domain spanned by itself and the scaling 22 center (Fig. 1). 23 24 25 2.1.1 Derivation based on the virtual work approach 26 27 The scaled boundary transformation of geometry, wherein Cartesian coordinates (x,y) are 28 mapped to corresponding scaled boundary coordinates (ξ,η) is provided below. It states 29 that any point in the domain can be equivalently expressed based on the position of the 30 scaling center x0, some scaling factor in radial direction ξ and an interpolation in tangential 31 direction x(η). 32 33 x(ξ,η) = xO + ξx(η) = xO + ξ[N(n)] x (2a) 34 { } y(ξ,η) = yO + ξy(η) = yO + ξ[N(n)] y (2b) 35 { } 36 37 where [N(η)] corresponds to the ﬁnite element interpolation functions [N1(η),N2(η),...,Nn(η)], x x x x T n 38 represents the nodal coordinates [ 1, 2,..., n] and is the number of nodes per element{ } on the boundary. The Jacobian J, its inverse J 1 and determinant J , which link the 39 − derivatives in Cartesian to scaled boundary coordinates on the boundary are| | given as: 40 41 42 ( ∂ )  ∂x ∂y ( ∂ ) ( ∂ ) x x 43 ∂ξ = ∂ξ ∂ξ ∂ = [J] ∂ (3a) ∂  ∂x ∂y  ∂ ∂ 44 ∂η ∂η ∂η ∂y ∂y 45 ( ∂ ) " ∂y ∂y #( ∂ ) ( ∂ ) 46 ∂x 1 ∂η ∂ξ ∂ξ 1 ∂ξ = − = [J]− (3b) 47 ∂ J ∂x ∂x ∂ ∂ ∂y | | ∂η ∂ξ ∂η ∂η 48 − 49 As a result, the differential volume unit dV is expressed as follows in scaled boundary 50 coordinates: 51 52 dV = J ξdξdη (4) 53 | | 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 7 4 5 Subsequently, these derivatives are used to evaluate the linear differential operator L in 6 scaled boundary coordinates. Splitting L into Cartesian components facilitates the substitu- 7 tion of their scaled boundary equivalents. Factoring into scaled boundary components, this 8 results in: 9 10 1 0 0 0 ∂ ∂ ∂ ∂ 11 [L] = 0 0 + 0 1 = [L1] + [L2] (5a)   x   y x y 12 0 1 ∂ 1 0 ∂ ∂ ∂ 13 1 ∂y ∂ ∂y ∂ ∂x ∂ ∂x ∂ 14 = [L1] + [L2] (5b) J ∂η ∂ξ − ∂ξ ∂η ∂ξ ∂ξ − ∂η ∂ξ 15 | | ∂ 1 ∂ 16 =[b1(η)] + [b2(η)] (5c) 17 ∂ξ ξ ∂η

18 1 2 19 where [b (η)] and [b (η)] represent: 20 For Peer ∂y 0 Review 21 1 ∂η [b1(η)] =  0 ∂x  (6a) 22 J  − ∂η  23 | | ∂x ∂y − ∂η ∂η 24  ∂y 0 25 1 − ∂ξ [b2(η)] =  0 ∂x  (6b) 26 J  ∂ξ  | | ∂x ∂y 27 ∂ξ ∂ξ 28 − 29 The displacements u(ξ,η) follow an isoparametric formulation. They too consist of { } 30 an analytical part in radial direction u(ξ) and an interpolatory part in tangential direction { } 31 based on shape functions [Nu(η)]. 32 u 33 u(ξ,η) = [N (η)] u(ξ) (7) { } { } 34 where [Nu(η)] represents the shape functions [N(η)], which are applied to each DOF of an 35 element separately by means of multiplication with the identity matrix [I]: 36 37 u 1 2 n [N (η)] = [N (η)In,N (η)In,...,N (η)In] (8) 38 39 As a result, an expression for the strains may be derived by substituting the formulation for 40 the displacements into Eqn. 1c. 41 1 1 2 42 ε(ξ,η) = [B (η)] u(ξ) ,ξ + [B (η)] u(ξ) (9) { } { } ξ { } 43 44 where

45 1 1 u 46 [B (η)] = [b (η)][N (η)] (10a) 2 2 u 47 [B (η)] = [b (η)][N (η)],η (10b) 48 49 Substituting Eqn. 9 into the constitutive relation 1b results in the following expression for 50 the stress ﬁeld. 51 1 σ(ξ,η) = [D] [B1(η)] u(ξ) + [B2(η)] u(ξ) (11) 52 { } { },ξ ξ { } 53 54 55 56 57 58 59 60

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1 2 3 8 Adrian Egger et al. 4 5 In what follows, the virtual work method is implemented. To this end, the virtual displace- 6 ments δu(ξ,η) and strains δε(ξ,η) are expressed as follows: { } { } 7 δu(ξ,η) = [N(η)] δu(ξ) (12a) 8 { } { } 9 δε(ξ,η) = [L] δu(ξ,η) (12b) { } { } 10 Thus, by equating internal work to external work, the weak form of the scaled boundary 11 finite element method is derived in the following form: 12 13 Z Z δε(ξ,η) T σ(ξ,η) dV δu(η) T t(η) = 0 (13) 14 V { } { } − ∂Ω { } { } 15 P 16 The second term in Eqn. 13, gives rise to equivalent nodal loads , due to applied tractions t . The corresponding displacements calculated on the boundary{ } are termed u . Equating 17 (η) h the internal virtual work to the external virtual work statements we can thus formulate the 18 complete virtual work equation: 19 20 T 0 1 ForT Peer Review δu [E ] uh + [E ] uh P 21 { } { { },ξ { } − { }} Z ξ=1 22 T n 0 δu(ξ) [E ]ξ u(ξ) ,ξξ 23 − xi=0 { } { } 24 1 o + [[E0] + [E1]T [E1]] u(ξ) [E2] u(ξ) dξ = 0 (14) 25 − { },ξ − ξ { } { } 26 27 A detailed derivation can be found in the appendix, where the following coefficient matrices, which bare striking similarity to stiffness matrices in FEM, are introduced. The chosen 28 approach follows [23] closely. 29 30 Z [E0] = [B1(η)]T [D][B1(η)] J dη (15a) 31 ∂Ω | | 32 Z [E1] = [B1(η)]T [D][B2(η)] J dη (15b) 33 ∂Ω | | 34 Z [E2] = [B2(η)]T [D][B2(η)] J dη (15c) 35 ∂Ω | | 36 37 In order for this equation to hold for all ξ, which implies it should be continuously satisfied 38 in radial direction and only compliant in the finite element sense in the tangential direction, 39 the conditions specified by both Eqn. 16 on the boundary and Eqn. 17 in the domain, must 40 be satisfied: P = [E0] u + [E1]T u (16) 41 { } { },ξ { } 42 0 2 0 1 T 1 43 [E ]ξ u(ξ) ,ξξ + [[E ] + [E ] [E ]]ξ u(ξ) ,ξ 44 { } − { } [E2] u(ξ) = 0 (17) 45 − { } { } 46 The above equation is termed the scaled boundary finite element equation in displacement. 47 Alternatively, one could also formulate the scaled boundary finite element equation in the 48 time or frequency domain by including inertial effects into the virtual work equation. Eqn. 49 17 is equivalent to earlier work from Wolf and Song, who first used a mechanically based 50 approach [21] and then a weighted residual method [22]. However, this derivation is more 51 consistent with standard engineering principles and understanding. Eqn. 17 is characterized 52 as a homogeneous set of Euler-Cauchy differential equations. 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 9 4 5 6 λ = 0 λ = 0 λ = 0.5 λ = 0.5 7 − − 8 9 10 11 λ = 1 λ = 1 λ = 1 λ = 1 12 − − − − 13 14 15 Fig. 3 Modal interpretation of the general solution of SBFEM [54]. Dashed line: original domain; solid line: 16 displacement mode. 17 18 2.1.2 Solution procedure 19 20 The general solution to the scaledFor boundary Peer ﬁnite element equationReview can be written in power 21 series form as: 22

23 λ1 λ2 λn u(ξ) = c1ξ − φ1 + c2ξ − φ2 + ... + cnξ − φn 24 { } { } { } { } λ 25 = [φ]ξ d− c c (18) { } 26 where λi and φi are the corresponding eigenvalues and vectors respectively. The bound- 27 { } 28 ary conditions determine the integration constants ci. As a result, the solution of SBFEM in 29 statics strongly resembles the mode superposition method of FEM. Hence, the eigenvectors 30 φi can be interpreted as modal displacement vectors on the boundary with a corresponding{ } scaling factor in the radial direction (Fig. 3). A quadratic eigenproblem results from 31 λi substituting the general solution into the scaled boundary finite element equation in dis- 32 placements: 33 34 [ λ 2[E0] λ [[E1]T [E1]] [E2]] φ = 0 (19a) 35 d c − d c − − { } { } q = [[E1]T λ [E0]] φ (19b) 36 { } − d c { } 37 In Eqn. 19b the boundary forces are expressed in an equivalent modal formulation. Con- 38 ceptually, they constitute the nodal force modes q , which balance the corresponding dis- 39 placement modes on the boundary, who are directly{ } dependent on φ by definition of the 40 general solution. Linearizing the quadratic eigen-problem has been{ shown} to be beneficial 41 in the context of fracture mechanics [28]. Unfortunately, this means doubling the size of the 42 eigen-problem. This is achieved by rearranging the equivalent nodal forces equation (Eqn. 43 19b) to obtain: 44 0 1 1 T λ φ = [E ]− [[E ] φ q ] (20) 45 d c{ } { } − { } 46 which is in turn substituted into the scaled boundary finite element equation (Eqn. 19a): 47 0 0 1 1 T 1 T 48 λ [E ][E ]− [[E ] φ q ] λ [E ] φ d c { } − { } − d c { } 1 0 1 1 T 2 49 + [E ][E ]− [[E ] φ q [E ] φ ] = 0 50 { } − { } − { } { } which is equivalent to: 51 1 0 1 1 T 2 52 λ q = [E ][E ]− [[E ] φ q [E ] φ ] (21) d c{ } { } − { } − { } 53 54 55 56 57 58 59 60

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1 2 3 10 Adrian Egger et al. 4 λ 5 ξ 6 bounded 7 ∞ domain modes 8 unbounded 9 1 10 domain modes 11 12 0 ξ 13 0 1 ∞ 14 Fig. 4 Contributions of the domain modes to the bounded and unbounded domain solutions 15 16 17 The linearized, combined form of the quadratic eigen-problem can thus be expressed more 18 compactly in matrix notation as: 19 ( ) ( ) 20 For φ Peerφ Review 21 [Z] = λ (22) 22 q d c q 23 24 where " 0 1 1 T 0 1 # 25 [E ]− [E ] [E ]− [Z] = − (23) 1 0 1 1 T 2 1 0 1 26 [E ][E ]− [E ] [E ] [E ][E ]− 27 − − 28 where [Z] is a Hamiltonian matrix. This special matrix form mandates symmetry about the 29 real and the imaginary axis for all eigenvalues. As a result, the eigenvalues corresponding 30 to rigid body translational modes can pose significant numerical issues. Machine precision 31 can cause these eigenvalues to arbitrarily alternate in sign. Attempts by other authors to 32 compensate for this issue resulted in: 33 1. Minimizing the condition number by 34 – Preconditioning coefficient matrices [55] 35 – Normalizing domain properties [21] 36 2. Addition of a mesh and platform dependent stabilizing parameter ε to [E2]. The optimal 37 value of this parameter is unknown a priori ([21], [27]). This corresponds to physically 38 adding small springs to the boundary. 39 40 Since eigenvalues of opposite sign contribute to the response of the system in funda- 41 mentally different ways, which will be detailed in the following paragraph, they should be 42 sorted accordingly. 43 ( ) ( ) " #" # 44 φ φ λneg [φ1][φ2] [Z] = λ = d c (24) 45 q d c q λpos [Q1][Q2] 46 d c 47 The subscripts “pos” and “neg” refer to positive and negative eigenvalues respectively, which 48 bear the effect of separating the bounded from the unbounded response. For negative eigen- 49 values the contribution at ξ = 0 is bounded and at ξ = 1 has a finite value, which represents 50 a solution in agreement with that of the bounded domain, whereas for positive eigenvalues 51 the contribution at ξ = 1 is finite and at ξ = ∞ is bounded, resembling the solution of an 52 unbounded domain (Fig. 4). 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 11 4 5 Having determined the eigenvalues and eigenvectors, these can be substituted into the gen- 6 eral solution (Eqn. 18) for determining the domain stiffness matrix. 7 λneg λpos 8 u(ξ) = [φ1]ξ d− c c1 + [φ2]ξ d− c c2 (25) { } { } { } 9 Considering that only the negative eigenvalues contribute to the response of the bounded 10 domain, the terms associated with the positive eigenvalues of Eqn. 25 are discarded. Next, 11 in order to evaluate the stiffness matrix, the displacement modes must be linked to the force 12 modes. This is achieved by comparing the displacements on the boundary u(ξ = 1) with 13 the acting nodal forces P . Enforcing the corresponding boundary conditions,{ by applying} 14 the integration constants{ to} the force modes, results in the following formulation: 15

16 Pbounded = [q1] c1 (26) 17 { } { } 18 The integration constants are obtained by evaluating Eqn. 25 at ξ = 1. 19 1 c1 = [Φ1]− u(ξ = 1) (27) 20 For{ } Peer{ } Review 21 In a subsequent step, the integration constants are substituted into Eqn. 19b: 22 1 23 Pbounded = [q1][Φ1]− u(ξ = 1) (28) 24 { } { } 25 Therefore, the stiffness matrix of a bounded domain is found to be: 26 K = +[q ][Φ ] 1 (29) 27 bounded 1 1 − 28 The stiffness matrix is symmetric, though fully populated. Hence, the use of higher order 29 elements is not penalized by higher bandwidth as in standard FEM approaches. 30 Similarly, the stiffness matrix for the unbounded domain may be determined by utilizing 31 the analogous expressions of the unbounded domain: 32 1 Kunbounded = [q2][Φ2]− (30) 33 − 34 In order to perform the back calculation of strains and stresses, first the integration constants 35 (Eqn. 27) must be determined, next the general solution (Eqn. 25) is sought, and only then 36 are the strains (Eqn. 9) and stresses (Eqn. 11) derived. 37 In order to to better understand the steps, computational complexity and effort required 38 for an SBFEM analysis the flowchart in Fig. 5 is offered. Restricting discretization along 39 the boundary when using SBFEM comes mainly at the price of the solution of an eigen- 40 problem. Therefore, domains with minimal surface to volume ratio are especially suited for 41 analysis using SBFEM. 42 Very similarly to standard FE-based approaches, first element-wise coefficient matrices 43 are evaluated and assembled using a standard numerical integration procedure. However, this 44 must be performed for three separate coefficient matrices. As opposed to XFEM though, no 45 additional enrichment terms must be included, and thus the need for integration of singu- 46 lar terms is eliminated. Since in SBFEM all elements can be treated equally, many issues 47 pertaining to the condition number of the stiffness matrix can be avoided. Additionally, no 48 further assumptions, for instance pertaining to what type of singularity to include in the 49 crack tip enrichment, are put in place. 50 In 2D, since the domain is represented by a linked series of bar-type elements, the as- 51 sembled coefficient matrices are highly sparse and their entries are clustered about the di- 52 agonals facilitating the inversion required for constructing the Hamiltonian matrix [Z]. The 53 54 55 56 57 58 59 60

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1 2 3 12 Adrian Egger et al. 4 5 Phase SBFEM steps Eqn. #

6 input processing 7

8 pre- processing for each subdomain 9 10 for each element 11 for each integration point 10a b 12 −

evaluate coefﬁcient matrices 14a c 13 −

14 assemble coefﬁcient matrices 15 16 form Hamiltonian 23 solution

17 Eigen decomposition 24 18 19 stiffness matrix 29

20 domain stiffness matrixFor Peer Review 21 22 apply boundary conditions

1 23 solve FEM equation U K− F = 24 27 25 back calculations 25 9,11

26 post- processing 27 28 Fig. 5 Flowchart of a SBFEM analysis linked to corresponding equations from section 2.1. 29 30 eigen-decomposition is the central core of SBFEM analysis, as it not only is necessary for 31 determining the stiffness matrix, but also delivers the analytical solution for the displace- 32 ments in radial direction. Once the stiffness matrix is formed, the remaining steps coincide 33 with those of the standard FEM approach. Significant differences may be identified, in the 34 calculation of the SIFs, when compared against the XFEM approach. Not only are the SIF 35 easily identified, efficiently extracted in post-processing, smoothed for further accuracy and 36 used to determine an effective error-estimator in SBFEM (section 2.3), but they are addi- 37 tionally exempt from parameter-specific dependencies, such as the radius of integration and 38 enrichment in XFEM. To see how this contributes to reducing computational complexity, 39 approximate flop counts are listed in Table 6. 40 41 42 2.2 Calculation of stress intensity factors 43 44 What sets SBFEM apart from other numerical methods is how elegantly highly accurate 45 SIFs may be obtained. The standard solution procedure by default accommodates cracks. 46 Values for the SIFs can easily be extracted during post-processing of the stresses with only 47 negligible additional computational effort. This is a consequence of the analytical solution in 48 radial direction. If the scaling center is placed at the crack tip, an analytical solution for the 49 strains and therefore additionally the stresses in radial direction may be computed. The SIFs 50 can be determined by evaluating the limit of the stresses as ξ 0. After substituting the 51 general solution for displacements into the definition of the stresses→ (Eqn. 11), the stresses 52 for any particular element can be obtained as a superposition of stress modes: 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 13

4 i

5 i+1

6 1 element 7 i+2 r y

L i+1 ( 8 θ) θ x 9 A i+3 lmn 2 element

10 Lθ=0 i+2 11 12 i+1 i+2 13 14 Fig. 6 Cracked SBFEM domain with crack tip location coinciding with scaling center. Additionally, the 15 DOFs of the two element used to compute the SIFs are identiﬁed. 16 17 18 n λi 1 19 σ(ξ,η) = ∑ ciξ − − Γi (31) 20 For Peeri=1 Review 21 22 with Γi representing the associated stress modes. 23   24 Γxx 25 Γi = Γyy = D[ λiB1(η) + B2(η)]Φi (32)   − 26 Γxy i 27 28 The contribution of each mode to the stress response is illustrated in Fig. 7. Only the modes 29 corresponding to eigenvalues within the range 1 < λ < 0 contribute to the singular re- − 30 sponse at the crack tip (ξ = 0). Consequently, for the purpose of determining the SIFs, all 31 other modes can be discarded. In 2D elastostatics, only two eigenvalues and thus modes 32 in the aforementioned range exist. As each element contains these singular modes, theo- 33 retically the SIFs could be extracted from any element in the domain. Here, the elements 34 lying directly along the crack path extension (Fig. 6 point A) are chosen. Once the singu- 35 lar modes have been identiﬁed, corresponding stresses can be theoretically computed using 36 values from anywhere within the bounded domain 0 < ξ < 1. Due to numerical concerns, 37 the singular stresses are evaluated at the boundary (ξ = 1) and consequently the power ex- 38 pression for ξ in Eqn. 31 is reduced to 1. 39 40 Contribution of eigenvalues to stresses 41 5

42 6 = 0.5 4 6 = 3 43 6 = 5 6 = 7

44 3 6

45 9 46 2 47 1 48 49 0 0 0.2 0.4 0.6 0.8 1 1.2 50 9 51 Fig. 7 Contribution of eigenvalues to the stress solution, reafﬁrming that 1 < λ < 0 leads to singularities 52 as ξ 0. − → 53 54 55 56 57 58 59 60

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1 2 3 14 Adrian Egger et al. 4 5 Since the singular components of the stresses can be derived analytically, these can be 6 matched directly to the definition of the SIFs. The integration constants ci correct for the 7 boundary conditions. L0 is then defined as the distance from the crack tip to the location of 8 the evaluated stresses. Thus, the mode I and mode II SIFs assume the following form: 9 10 KI p ∑i=I,II ciΓyy(η = ηA)i = 2πL0 (33) 11 KII ∑i=I,II ciΓxy(η = ηA)i 12 13 The stress modes Γi need only be evaluated at one specific point ( 1 < η = ηA < 1) along 14 − the element. In order to accomplish this, the correct submatrix φi must be identified. This 15 consists in finding the eigenvector linked to the singular eigenvalue, as well as identifying 16 the rows representing the DOFs of the corresponding element on the boundary. Their inter- 17 section (Fig. 8) marks the relevant data for the calculation of the SIF modes. Consequently, 18 the calculation of the SIFs requires only minimal computational effort. 19 20 For Peer Review 21 λ1 22 0.5 φ φ φ 23 1.,1 . 1.,j . 1.,n . ···. . ···. . DOF 24 . . . . . 25 φi,1 φi,j φi,n

. .··· . .··· . [ . . . . . i,i 26 [φ] = . . . . . +5] 27 φi+5,1 φi+5,j φi+5,n . .···. . .···. . 28 . . . . . φn,1 φn,j φn,n 29 ··· ··· 30 Fig. 8 Singular displacement modes as columns and associated DOFs in rows of a representative element on 31 the boundary. The dark gray intersection signifies the submatrix φi required in order to compute the SIFs. 32 33 34 35 2.3 Recovered stress intensity factors and error estimator 36 37 Since the SIFs are determined solely on the basis of calculated stresses, they can be easily 38 inferred. Should the crack extension fall directly between two elements, different stresses are 39 calculated for each element at the respective node A (Fig. 9). Trivial averaging of the stresses 40 is an option, but not necessarily the best one. Thus, in this work, a novel application of SPR 41 theory to SIFs is proposed, resulting in superconvergent and in select cases ultraconvergent 42 convergence rates for the SIFs. This extends previous work by Deeks [34]. 43 Recovered stresses are gained by least squares fit of the raw stresses sampled at the 44 Gauss integration points over a patch of two elements. For the case of enhancing SIFs, the 45 two elements are proposed to coincide with those in the direction of crack extension. Based 46 on [34] and [22] the fitted recovered stresses are calculated as: 47 48 σ rec(η) = P a (34) 49 { }{ } 50 51 where 1 < η < 3 (Fig. 9) and for this specific application η = 1, i.e., point A. P is a − { } 52 vector of powers of η and a is a vector of undetermined coefficients ai. The vector of raw 53 { } 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 15 4 5 stresses computed at the Gauss integration points is denoted by σ raw. 6  0T    raw 7 η a1 σ1  1    raw 8 η  a2 σ2        9  .   .   .   .   .  raw  .  10 P = i , a = ,σ = raw (35) { } η  { } ai  σi  11  .   .   .   .   .   .  12  .   .   .   n    raw 13 η an σn 14 15 where n denotes the amount of Gauss integration points present in the patch. The vector 16 of unknown coefficients a can be determined by evaluating vector P at each Gauss { } ¯ { } 17 integration point, and adding its contribution to matrix [P]. 18  0 2 i n η1 η1 ... η1 ... η1 19 0 2 i n η2 η2 ... η2 ... η2  20 For Peer Review  ......  21  ......  [P¯] =   (36) 22 η0 η2 ... ηi ... ηn  i i i i  23  ......   ......  24 0 2 i n 25 ηn ηn ... ηn ... ηn 26 The least squares problem can then be formulated as: 27 28 [P¯]( σ raw [P¯] a ) = 0 (37) 29 { } − { } 30 Subsequently, the vector of unknown coefficients is determined as: 31 T 1 T raw 32 a = ([P¯] [P¯])− [P¯] σ (38) { } { } 33 34 Thus, recovered stresses in the direction of crack extension are determined by substituting 35 η = ηA (Fig. 9, point A) into Eqn. 34. These stresses are indirectly recovered, by smoothing 36 the stress modes Γi (Eqn. 32), which in turn rely on the displacement modes [Φi]. Since 37 only two displacement modes contribute to the singular response, only a small subset of the 38 eigenmodes resulting from the SBFEM solution must be smoothed. This is the first means used to reduce the computational effort for determining the SIFs. Further, since only one 39 patch in direction of crack extension is considered, recovery over only two elements, again 40 a small subset of the whole domain, is necessary. It is assumed that this trade off in potential 41 accuracy is offset by the gain in computational speed. This is later demonstrated to hold true 42 in section 3. Consequently, Eqn. 31 may be rewritten as: 43 44 2 s λi 1 s 45 σ (ξ,η) = ∑ ciξ − − Γp (39) 46 i=1 47 in which superscript s denotes the singular modes and subscript p corresponds to the stress 48 modes smoothed on the patch in direction of crack extension. Since the stresses are evalu- 49 ated at the boundary, ξ = 1 and Eqn. 39 simplifies to the extent that the SIFs can now be 50 formulated as: 51 rec 2 s KI p ∑i=1 ciΓp,yy(η = ηA) 52 rec = 2πL0 2 s (40) KII ∑i=1 ciΓp,xy(η = ηA) 53 54 55 56 57 58 59 60

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1 2 3 16 Adrian Egger et al. 4 5 Finally, it stands by virtue of the SPR theory, that the so-recovered SIFs should be at 6 least superconvergent. 7 Based on the difference in raw and recovered stresses, a novel error estimator accompa- 8 nying the proposed recovery scheme for SIFs can be deduced. 9 e∗ (ξ,η) = σ ∗(ξ,η) σh(ξ,η) (41) 10 { σ } { } − { } 11 where superscript * denotes the recovered value and subscript h denotes the raw value of 12 the stress calculated in direction of crack extension. By consequence, obtaining the error 13 estimator is rendered computationally inexpensive. 14 15 16 Γi 17 18 η 19 element 1 A element 2 node raw stresses 20 Forintegration point Peer Review recovererd stresses 21 Fig. 9 Stress recovery based on SPR. A least square fit is constructed for the stresses in point A, based on 22 values computed at the Gaussian integration points. 23 24 25 26 27 28 2.4 Block Hamiltonian Schur Decomposition 29 30 The solution process in SBFEM benefits from the HSchur decomposition in multiple ways. 31 – It eliminates the need for an arbitrary stabilizing parameter ε. 32 – By respecting Hamiltonian symmetry of the eigenvalues, regular and steeper conver- 33 gence rates can be achieved. 34 – The doubling of the problem size when linearizing the quadratic eigenvalue problem is 35 reversed and thus computational complexity is significantly reduced. 36 – It still benefits from a Schur form, which allows to quantify general SIFs for bi-material 37 interfaces. 38 The following is an abridged treatment of the theory first presented in [26] with emphasis 39 placed on the parts integral to SBFEM. 40 2n 2n A matrix H R × is Hamiltonian iff: 41 ∈ 42 AG H = (42) 43 start C AT 44 − 45 Additionally, GT = G and CT = C, which holds for the case of SBFEM (Eqn. 23). 46 Matrix H is said to be in Hamiltonian real Schur form if 47 TG 48 H = . (43) final 0 T T 49 − 50 where T is quasitriangular. Any eigenvalue +λ found in T has a corresponding eigenvalue 51 λ in T T . For most applications it makes sense to group eigenvalues with same sign into 52 T− and −T T respectively. 53 − 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 17 4 5 In order to arrive at Hfinal from Hstart , a series of orthogonal symplectic similarity trans- 6 formations are performed: T 7 Hi+1 Qˆ HiQˆ (44) ← 8 Every step is accompanied by the corresponding update, for which the starting values of Q0 9 is chosen as I2n: 10 Qi+1 QiQˆ (45) 11 ← 12 In algorithmically formulating this process of arriving at the Hamiltonian real Schur form, 13 first a symplectic URV decomposition of H is performed. The SLICOT library [56] provides 14 the necessary backwards stable routines: 15 SG 16 H = UR V T with R = (46) 1 1 0 T T 17 18 n n n n n n T R × is upper-triangular, S R × is quasitriangular and G R × . It is at this point 19 that∈ the eigenvalues of H can be∈ determined as they are equal to∈ the square roots of the 20 eigenvalues of the quasitriangularFor matrix PeerST. Review 21 Next, the eigenvalues of ST, which correspond− to the eigenvalues of H2 are partitioned 22 into blocks. Many clustering− algorithms exist, however here we choose an open ball with 23 radius: 24 r = 10 S F T F κ ε (47) 25 × || || || || × × F denotes the Frobenius norm, κ is the condition number of the eigenvalue, ε is machine 26 ||•|| 27 precision and the factor 10 is included to arrive at a conservative estimate as originally 28 proposed in [26]. Eigenvalues of overlapping balls are placed in the same block. Complex 29 conjugates are then added to the block if they exist as well. In a last operation for this 30 step, the eigenvalues of the URV decomposition are ordered to match their blocks. The new structure of the matrix ST is given as follows: 31 − 32   B11 ... B1s 33 . ST = B =  .. .  (48) 34 −  . .  35 0 Bss 36 The spectrum of B is given by eigenvalues Λ of block 1 i s. At this point, the first 37 ii i < < orthogonal symplectic update is performed using the sorted output of the URV decomposi- 38 tion: 39 H UT HU (49) 40 ← 41 Henceforth, each block will be considered separately. The general procedure will be illus- 42 trated based on the first block B11 with size k. First, a QR decomposition is performed on: 43 44 [Ek HEk] = QR (50) 45 E 2n k is the submatrix of I constructed from the first k columns. Next, a Schur 46 k R × 2n decomposition∈ is performed: 47 48 T Schur T G = Q HQ QeTeQe (51) 49 −−−→ 50 The eigenvalues of Qe are subsequently sorted into left- (Qe1) and right (Qe2) half plane re- 51 spectively: 52 Qe = [Qe1,Qe1] (52) 53 54 55 56 57 58 59 60

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1 2 3 18 Adrian Egger et al. 4 5 In order to construct an orthonormal basis for the invariant subspace of H, a matrix X is 6 introduced as follows: 7 X = QQe1 (53)

8 The main task lies in transforming X into Ek and thus achieve a deflation. This is carried T 9 out by means of finding an orthogonal symplectic S such that S X = Ek, which will be 10 subsequently used to perform an update on H ST HS. To this end, first X is partitioned to 11 conform to the block sizes: ← 12  X1  13  .   .  14    Xs  15 X =   (54) Xs 1 16  +   .  17  .  18 X2s 19 S can be found as: 20 For Peer Review 21 Sˆ 0 S = with Sˆ = diag Sˇ,I and Sˇ = QFe (55) 22 0 Sˆ { } 23 24 Fe is a flip vector obtained from reversing the column order of the identity matrix [I], Q 25 results from the QR decomposition of the submatrix of X corresponding to [Xs+1;Xs+2]. After carrying out the updates X SeT X and H SeT HSe, the vector X assumes the following 26 ← ← 27 form:  X  28 1  .  29  .    30  Xs    31 X =  0  (56)   32 Xs 2  +  33  .  34  .  35 X2s 36 Subsequent iterations on the lower blocks of X, introduce further blocks of zeros eventually 37 resulting in:   38 X1 39  .   .  40    Xs  41   X =  0  (57) 42    .  43  .    44  0  45 X2s 46 For the last block a slightly modified procedure must be applied as there is no successive 47 block, which could be used to install a further block of zeros. For the last block the actions 48 are modified, by considering the submatrix [Xs;X2s]. Based on the QR decomposition of 49 T the aforementioned submatrix, from which Qˇ results, the required updates X Qe X and ← 50 H QeT HQe are constructed: 51 ← Xs Qˇ1 52 = R (58) X2s Qˇ2 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 19 4 5 and 6 Qe1 Qe2 ˇ ˇ 7 Qe = − and Qe1 = diag I,Q1 ; Qe2 = diag I,Q2 (59) Qe2 Qe1 { } { } 8 9 This results in the entire lower portion of X running from blocks s + 1 < k < 2s to be filled 10 with zeros. 11 In a next step, an analogous procedure for the block 1 < k < s will be applied. S can be 12 found as: 13 ˆ 14 S 0 ˆ ˇ ˇ S = ˆ with S = diag S,I and S = Q (60) 15 0 S { } 16 Q results from the QR decomposition of the submatrix of X corresponding to [Xs 1;Xs]. 17 − Again a block of zeros is inserted into X: 18 19   20 For PeerX1 Review  .  21  .  22   Xs 10 X =  −  (61) 23  0  24    .  25  .  26 0 27 28 This procedure is continued until only one block remains. After 2s 1 updates of X SeT X − ← 29 and H SeT HSe the desired deflation is nearly achieved. At this point, the Hamiltonian 30 matrix H←is of the form: 31 A A G G  32 11 12 11 12  0 A22 G21 G22 33 H =  T  (62)  0 0 A11 0 34 − T T 0 C22 A A 35 − 11 − 22 36 It is now admissible to deflate the problem. Subsequent operations are performed on the 37 following Hamiltonian submatrix: 38 39 A22 G22 40 H = T (63) C22 A 41 − 22 42 43 This deflation is repeated for each block until the real Hamiltonian Schur form is reached: 44 45 AG H = (64) 46 0 AT − 47 48 Q contains the orthonormal basis for the stable invariant subspace. In a last step, a sorting of 49 the eigenvalues and vectors spanning the stable invariant subspace is performed, in order to 50 retain only negative eigenvalues. Such routines are also available in the Slicot library [56]. 51 It therefore becomes evident that application of the HSchur decomposition in the solu- 52 tion process of SBFEM as proposed, will increase accuracy, while reducing complexity. 53 54 55 56 57 58 59 60

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1 2 3 20 Adrian Egger et al. 4 5 3 Numerical Examples 6 7 The first numerical example illustrates the benefits to the overall robustness of the pro- 8 posed SBFEM solution, incorporating the HSchur decomposition, versus the established 9 eigen and standard Schur decomposition. This novel contribution alleviates the need for 10 an architecture-dependent stabilizing parameter ε. The second numerical example demon- 11 strates the consequently improved convergence behavior of the L2 displacement and stress 12 norms. Additionally, an efficient procedure is proposed to enhance the accuracy of the cal- 13 culated SIFs. According to the latter, only local stress recovery over select few elements is 14 performed on the boundary instead of over the whole domain, simultaneously resulting in 15 significantly reduced computational effort. Further, it is established that the SIFs calculated 16 by the proposed method are superconvergent for even-noded elements or even ultraconver- 17 gent for some odd-noded cases. This forms the basis for a novel error estimator for SIFs. As 18 part of the third numerical example, the computational effort for an SBFEM solution is con- 19 trasted to that of FEM, based on a flop count, which is demonstrated for the first time herein. 20 Additionally, the enhancementFor in accuracy Peer for the proposed Review local recovery of the stresses 21 compared to adoption of trivial averaging of raw stresses at nodes is demonstrated. Next, 22 SBFEM is directly contrasted to XFEM and the FE-based commercial software Abaqus in terms of SIF reconstruction and conditioning of the stiffness matrix. To the authors knowl- 23 edge, this is the first occurrence of a direct comparison between standard SBFEM and XFEM 24 in literature. As XFEM represents a major competing method to SBFEM within the context 25 of LEFM, the fourth numerical example primarily considers this comparison. Furthermore, 26 the computational time required for various parts of the SBFEM procedure are shown, il- 27 lustrating the computational gains achieved by the proposed HSchur decomposition and 28 smoothing processes. 29 Calculations were performed on an Intel Core i5 6600K and an Intel Xeon E3-1225 v3. 30 SBFEM and XFEM where implemented in Matlab 2015b. In Abaqus 6.14-1 the contour 31 integral method was used to determine the SIFs. 32 33 34 3.1 Edge-cracked square plate under bending 35 36 A square, homogeneous domain with an edge crack as depicted in Fig. 10 is considered, 37 with length L = 2 units and crack length a = L/2. The domain boundary is uniformly dis- 38 cretized. For this special case of pure bending, an analytical solution exists for the SIFs [57]. 39 The application of point boundary restraints in SBFEM is similarly problematic as would 40 be in the standard FEM case, as these must be applied for eliminating rigid body motion. 41 Since in this numerical example restraints have to be positioned along the direction of crack 42 extension, they may induce restraining forces precisely at the position used to calculate the 43 SIFs. This complicates the accurate determination of the SIFs, though it may be mitigated 44 to an extent for this special test case by scaling the problem domain uniformly. This is ad- 45 missible since the SIF for an edge cracked square plate under bending only depends on the 46 ratio of side length to crack length [57]. 47 Numerical difficulties arise in the solution process of SBFEM due to the occurrence 48 of rigid body modes. In 2D elastostatics, a 4x4 block of zero eigenvalues exists. Two are 49 associated with the response of the bounded domain, whereas the other two contribute to 50 the response of the unbounded domain (Fig. 4). Numerical issues may perturb eigenvalues, 51 resulting in signs, which may alternate arbitrarily. A subsequent reordering of eigenvalues 52 into left half plane and right half plane may thus result in an eigenvalue intended for the 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 21 4 σ = 1 5 6 7 8 E = 200 [N/mm2] 9 ν = 0.3 L 10 a = L/2 11 12 13 14 15 16 L 17 Fig. 10 Edge-cracked square plate subject to bending. The problem statement for the first numerical example. 18 19 20 unbounded domain to falselyFor contribute toPeer the bounded domain. Review Since the stiffness matrix 21 is constructed from the corresponding eigenvectors, this effect leads to erroneous results. As 22 an example, it is possible that the rigid body mode of horizontal displacement is considered 23 twice, whereas the rigid body mode of vertical displacement is not considered at all and vice 24 versa (Fig. 3). 25 Since one is not able to predict the outcome of the sorting a priori, a stabilizing parameter 26 ε is typically introduced in existing literature [21], with the intent of shifting the eigenvalues 27 of the rigid body modes off of the imaginary axis. This is achieved by modifying the input 28 to the Hamiltonian Matrix [Z] (Eqn. 23) as follows, which physically bears the effect of 29 introducing springs on the boundary: 30 2 2 31 [E ] = [E ] + ε[I] (65) 32 Table 3.1 summarizes the solution quality of SBFEM for various values of epsilon. Ac- 33 cording to [21], the typical values for epsilon range from 10 6 < ε < 10 12. Below, the 34 − − eigen decomposition is considered, though similar effects may also be observed for the 35 standard Schur decomposition. It is evident, that values of ε exist, where incorrect solutions 36 are obtained. Further, there exist regions for which the choice of ε does not bear signif- 37 icant influence on the solution. Remarkable is the fact that on different CPUs, the values 38 for epsilon, which result in solution errors due to incorrect separation of eigenvalues, differ. 39 Consequently, the correct choice of epsilon cannot be determined prior to the analysis. In or- 40 der to ensure robust results, multiple runs with different values for epsilon are necessary and 41 usually require user intervention, which reduces the computational advantage of SBFEM. 42 This significantly complicates modeling of crack propagation and effectively bars SBFEM 43 from being used for problems, where successive function evaluations are required, such as 44 optimization problems using heuristic methods. 45 The main advantage of the HSchur decomposition stems from the fact that it accounts 46 for the special structure of the Hamiltonian Matrix [Z], which dictates symmetry about the 47 real and imaginary axis for all eigenvalues. Taking advantage of symmetry by only consid- 48 ering the left half plane associated with the negative eigenvalues, the eigenvalues are always 49 separated properly, and the need for a stabilizing parameter is removed. Thus, this novel 50 approach enables a robust, parameter independent solution process for SBFEM. 51 This symmetry may be observed in Fig. 11, where the eigenvalues for the square plate 52 are plotted. The HSchur decomposition better respects symmetry in comparison to the cor- 53 54 55 56 57 58 59 60

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1 2 3 22 Adrian Egger et al. 4 5 Table 1 Correct solutions obtained as a function of the stabilizing parameter ε for various CPU architectures. 6 Discretized with 3-noded elements and 6 elements per side. 7 () 8 ε = 10− 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 Core i5 X X X X X X X 9 X X X X X X X X X X X X Xeon E3 X X X X X X X X X X X X X X X X X X X 10 11 12 Hamiltonian eigenvalue symmetry error 13 14 3 imag 15 2 16 1 17 real

18 -10 -5 5 10 19 -1 20 -2 For Peer Review 21 Schur Eigen -3 22 HSchur 23 24 Fig. 11 Symmetry of eigenvalues for eigen, Schur and HSchur decomposition. Discretized with 2-noded 25 elements and 4 elements per side. 26 !12 27 HSchur 0 = 0 0 = 10 28 29 30 31 0 !10 0 !8 0 !6 32 = 10 = 10 = 10 1.2 33

34 1 35 36 0.8 37 38 Fig. 12 Symmetry of stiffness matrix for various choices of ε compared to HSchur decomposition. Dis- cretized with 2-noded elements and 4 elements per side. 39 40 41 responding eigen and Schur decompositions. For the Schur decomposition, the block diago- 42 nalization seems to weaken symmetry. The eigen-decomposition approach does not require 43 this step, resulting in improved symmetry. However, the issues pertaining to the sorting of 44 the rigid body modes remains. Slight variation of the stabilizing parameter induces variation 45 in the eigenvalues, resulting in modiﬁed eigenvectors. As a result, the stiffness matrix loses 46 its symmetry (Fig. 12), which results in solution errors. 47 A large value for the stabilizing parameter epsilon results in loss of solution accuracy, as 48 introducing springs on the boundary effectively results in a different structural system. On 49 the other hand, too small values for epsilon may result in incorrect solutions due to improper 50 separation of eigenvalues. Fig. 12 visualizes, that there is a region for ε, where the solution 51 is better behaved. For this purpose, the values in the stiffness matrix (Ki, j) are compared 52 to the averaged sum (Kave = (1/2) (Ki, j + Kj,i)). According to Brenner and Kressner [58] 53 ∗ 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 23 4 Condition number vs #DOF 5 10 9

6 Eigen, 0 = 12.5 HSchur 7 10 8 8 7 9 10

) 6

10 K 10

(

d n

11 o c 12 10 5 13 10 4 14

15 10 3 16 0 25 50 75 100 125 150 175 200 225 250 275 17 #DOF 18 Fig. 13 Evolution of the condition number for eigen and HSchur decomposition. Discretized with 2-noded 19 elements. 20 For Peer Review 21 22 this difficulty of computing the zero and negative eigenvalues of a Hamiltonian matrix is 23 well known for QR and Arnoldi based decompositions, leading to loss of symmetry due to 24 round-off errors. It is further well-known that accuracy is compromised, when computing 25 eigenvalues of a matrix whose elements differ by several orders of magnitude [59]. Since 26 this is the case for the entries of the Hamiltonian matrix [Z], due to the contribution of the 27 inverse of [E0], this further compounds numerical errors. Normalizing the contributions of 28 the coefficient matrices [21] or applying a preconditioner [55] mitigate this effect partially, 29 however the issues discussed in [58] still remain. 30 Further, the condition number of the stiffness matrix varies, depending on different 31 choices of stabilizing parameter. Fig. 13 depicts the evolution of the condition number for 12.5 32 the eigen-decomposition with ε = 10− compared to the HSchur decomposition. How- 33 ever, since the condition number additionally varies as a function of ε for the standard 34 eigen-decomposition case, box plots are further provided, indication variation for the range 6 15 35 of 10− < ε < 10− . The observed spread in condition number may pose unpredictable, nu- 36 merical challenges. This spread is not present for the proposed HSchur decomposition. The 37 same is observed for matrix [Φ] (Eqn. 29), whose inversion is required for the formation of 38 the stiffness matrix and the evaluation of the vector of coefficients [c] (Eqn. 27). 39 Since the HSchur decomposition results in a robust and predictable solution for SB- 40 FEM, the calculated SIFs result independently of the stabilizing parameter epsilon. Fig. 14 41 demonstrates the fluctuation in computed SIFs by eigen-decomposition, contrasted to the 42 estimates of the HSchur decomposition. It is observed that the error in calculated SIFs via 43 eigen-decomposition with a variable ε may outweigh the inherent accuracy of the method 44 itself. Additionally, as is later demonstrated in the next numerical example, the L2 displace- 45 ment error for lower order elements also fluctuates significantly with variation in ε (Fig. 20). 46 This is not the case when employing the HSchur decomposition (Fig. 21).

47 In [57] an exact value for K1 is provided for the edge cracked plate subjected to pure 48 bending. In Figs. 15 and 16 respectively, the SIF K1 computed using raw stress and recov- 49 ered stress is depicted. Stress recovery is performed only on the two element patch in the 50 direction of crack extension (Fig. 6), and not on the entire domain. Table 2 indicates that 51 for the proposed case of local recovery the computational effort is drastically reduced, while 52 achieving similar accuracy to full domain recovery. This is important, since stress recovery 53 54 55 56 57 58 59 60

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1 2 3 24 Adrian Egger et al. 4 Error in K1 for a range of 0 5 0.7 6 7 0.6

8 0.5

9 1

K 0.4 n

10 i

o r

11 r e 0.3

12 % 13 0.2 14 0.1 Error eigen 15 Error HSchur 16 0 -16 -14 -12 -10 -8 -6 17 0!() 18 Fig. 14 Error in SIF K1 computed by eigen-decomposition as a function of ε, contrasted to the HSchur 19 decomposition. Discretized with 3-noded elements and 6 elements per side. 20 For Peer Review 21 22 in SBFEM occupies a signiﬁcant portion of the total execution time (Table 5). The formula- 23 tion of an error estimator for SIFs based on the difference of the raw and recovered stresses, 24 as proposed herein, may be exploited to greatly facilitate the SBFEM post-processing pro- 25 cedure. In Fig. 17 the computed error estimator is contrasted to the exact error for the SIF, 26 where good agreement is noted. The proposed procedure allows for inherent checking of the 27 accuracy of the computed SIFs, within a single analysis and with minimal computational 28 effort. 29 30 Table 2 Percent of total analysis time spent on calculating K1 based on full or local domain stress recovery. 31 Discretized with 3-noded elements and 3 elements per side, 4-noded elements and 4 elements per side and 32 5-noded elements and 5 elements per side respectively. 33 #DOF full domain local domain 34 % total time % error K1 % total time % error K1 35 98 36.03 0.128 0.24 0.131 36 198 30.63 0.019 0.10 0.020 322 26.11 0.001 0.02 0.001 37 38 39 40 41 3.2 Edge-cracked square plate with forced SIFs 42 43 The square, homogeneous domain with an edge crack, as depicted in Fig. 18 is considered, 44 with length L = 2 units and crack length a = L/2. The boundary is uniformly discretized. 45 An analytical solution for displacements, stresses and SIFs is given in [60]. The expressions 46 linking the displacements, stresses and SIFs are given in Table 3. 47 For this example, the mode I and mode II SIFs are chosen as K1 = 0 and K2 = 1. The 48 coinciding exact displacements on the boundary are then enforced as boundary conditions 49 at each node. The improvements in convergence properties when using the novel approach 50 based on the HSchur decomposition are investigated. To this end, ﬁrst the L2 error in dis- 51 placements norm is considered, followed by the L2 error in stresses norm. Figs. 20 and 21 52 demonstrate the L2 error in displacements norm for the eigen and HSchur decomposition 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 25 4 Error in K1, averaged stress 5 101 6

7 100 8 10-1

9 1

K n

10 i -2

r 10

o r

11 r e

12 % 10-3 13 2-Nodes m = 2.02 3-Nodes m = 1.95

-4 4-Nodes m = 3.75 14 10 5-Nodes m = 4.47 15 6-Nodes m = 5.56 16 10-5 102 17 #DOF 18 Fig. 15 Error in SIF K computed using raw averaged stress. 19 1 20 For Peer Review Error in K1, recovered stress 21 101 22 0 23 10 24

10-1 1

25 K

n i -2

26 r 10

r r

27 e % 28 10-3 2-Nodes m = 1.98 29 3-Nodes m = 3.97 -4 4-Nodes m = 3.86 10 30 5-Nodes m = 4.87 31 6-Nodes m = 5.89 10-5 32 102 33 #DOF 34 Fig. 16 Error in SIF K1 computed using recovered stress. 35 36 Error estimator for K1 10 2 37 38 10 1 39 10 0

40 1

K 10 -1

41 n

o r

42 r 2-Nodes ext -2 e 10 3-Nodes ext 43 % 4-Nodes ext 5-Nodes ext 44 10 -3 6-Nodes ext 2-Nodes est 45 3-Nodes est -4 4-Nodes est 10 5-Nodes est 46 6-Nodes est

47 10 -5 10 2 48 #DOF 49 50 Fig. 17 Proposed error estimator for SIFs contrasted to exact error. 51 52 53 54 55 56 57 58 59 60

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1 2 3 26 Adrian Egger et al. 4 5 6 KII = 1 E = 200 [N/mm2] 7 ν = 0.3 L 8 a = L/2 9 10 11 12 L 13 Fig. 18 Edge-cracked square plate problem domain subject to K2 = 1 as boundary conditions. The problem statement for the second numerical example. 14 15 16 Table 3 Analytical solution for edge cracked plate under forced K 17 2 K2 p r 2 18 u1: 2 µ 2π sin(θ/2)(κ + 1 + 2cos (θ/2)) ∗ 19 K2 p r 2 u2: 2 µ 2π cos(θ/2)(κ 1 2sin (θ/2)) 20 − ∗ For− − Peer Review σ11: K2/√2πr sin(θ/2)(2 + 2cos(θ/2)cos(3θ/2)) 21 − 22 σ22: K2/√2πr sin(θ/2)cos(θ/2)cos(3θ/2)

23 σ12: K2/√2πr cos(θ/2)(1 2sin(θ/2)sin(3θ/2)) 24 − 25 26 27 28 29 30 31 ξ 32 O 33 34 Fig. 19 Points at which the L2 norms are calculated, denoted by crosses. The upper left quarter of the domain 35 is depicted. 36 37 38 respectively. Since the boundary conditions are applied to all nodes, the error norm is calcu- 39 lated based on chosen displacements post-processed at 10 equally spaced points along ξ, as 40 denoted by crosses in Fig. 19. The same procedure was carried out in the case of the stresses. 41 42 A signiﬁcant improvement in the regularity and rate of convergence is observed between 43 the proposed HSchur decomposition (Fig. 21) and the standard eigen-decomposition (Fig. 44 20), as optimal convergence is almost achieved. Since the proposed HSchur decomposition 45 is independent of a stabilizing parameter, its convergence properties do not exhibit oscil- 46 lations, as is observed for the standard eigen-decomposition. For the eigen-decomposition, 47 these oscillations may be amended, by appropriate variation of ε, a task, which however is 48 not straightforward. This further implies that an increase of the employed number of DOFs 49 does not guarantee higher accuracy in the eigen-decomposition case. Furthermore, the con- 50 vergence rates m of this newly proposed method, better approximate the values predicted 51 by established FEM theory. Optimal values mopt are equal to the order of the interpolation 52 function. For SBFEM, this is equal to the amount of nodes present in each element. 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 27 4 L2 displacement error, Eigen decomposition 5 10-4 6 7 10-5

8 10-6

9 r

o -7

r 10 r

10 e

s i

11 d -8

2 10 12 L

-9 13 10 2-Nodes m = 1.94 3-Nodes m = 3.93 14 4-Nodes m = 3.99 -10 10 5-Nodes m = 4.86 15 6-Nodes m = 5.71 16 10-11 102 17 #DOF 18 Fig. 20 Convergence results of the L displacement error based on the eigen-decomposition. 19 2 20 For Peer Review 21 L2 displcement error, HSchur decomposition 10-4 22 23 10-5 24 -6

25 10 r

o -7 r 10

26 r

p s 27 i d -8

2 10

28 L

-9 29 10 2-Nodes m = 2.02 3-Nodes m = 3.99 30 4-Nodes m = 3.96 -10 10 5-Nodes m = 4.96 31 6-Nodes m = 5.93

32 10-11 102 33 #DOF 34 35 Fig. 21 Convergence results of the L2 displacement error based on the HSchur decomposition. 36 37 38 Next, the performance of i) the Schur decomposition as provided in Matlab, ii) the 39 Hamiltonian Schur decomposition of [25], and iii) the proposed new blocked Hamiltonian 40 Schur decomposition, HSchur, is further investigated via the following residual metric [25]: 41 T U HU Hˆ 2 42 ResidualSchur = || − || (66) Hˆ 2 43 || || 44 where Hˆ represents the computed Hamiltonian Schur form and H denotes the original 45 Hamiltonian Matrix. The residual of the current Matlab implementation of the Schur decom- 46 position, does not vary signiﬁcantly with changing values of ε. In Table 4 the residual for all 47 three methods is provided for different levels of discretization. The newly proposed HSchur 48 decomposition clearly outperforms the competing methods, which results in improved con- 49 vergence rates and elimination of the oscillations of the standard eigen-decomposition (Fig. 50 20). It should be noted that the Hamiltonian Schur method (CSchur) proposed by Chu et al. 51 [25], which is the precursor to the currently proposed method, has previously been shown to 52 exhibit increased residuals when clusters of tightly grouped eigenvalues are present. 53 54 55 56 57 58 59 60

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1 2 3 28 Adrian Egger et al. 4 5 Hamiltonian eigenvalue symmetry error

6 imag 15 7 8 10

9 5

10 real 11 -100 100 12 -5 13 -10 14 15 -15 16 HSchur 17 Fig. 22 Eigenvalue clustering for domains with higher levels of discretization. Discretized with 5-noded 18 elements and 10 elements per side. 19 20 Table 4 Residuals for each decompositionFor type. DiscretizedPeer with 3-noded Review elements. 21 22 # DOF Schur CSchur HSchur 160 5.4 10 1 8.1 10 5 9.1 10 11 ∗ − ∗ − ∗ − 23 226 6.3 10 1 6.2 10 5 7.5 10 12 ∗ − ∗ − ∗ − 24 280 2.7 10 1 9.0 10 3 6.3 10 9 ∗ − ∗ − ∗ − 25 360 3.5 10 2 1.3 10 2 5.2 10 9 ∗ − ∗ − ∗ − 448 7.7 10 2 2.6 10 2 2.8 10 12 26 ∗ − ∗ − ∗ − 27 28 29 Clustering of eigenvalues is generally the case for SBFEM, which is demonstrated in 30 Fig. 11 for few DOFs and Fig. 22 for a typical amount of DOFs necessary for such an 31 analysis. Naturally, as the amount of DOFs increases the clustering issues further propagate, 32 and the residual of CSchur tends to accordingly increase. The block of zero eigenvalues 33 [26] still poses a computational challenge for the proposed HSchur method. The presence 34 of purely imaginary eigenvalues results in failure to find an invariant, isotropic subspace to 35 working precision. By relaxing the isotropy checks, a near real Schur form is nonetheless 36 obtained, thereby forfeiting numerical accuracy. The is observed for residual values greater 37 16 than machine precision (eps = 2.22 10− ). 38 ∗ Figs. 24 and 23 plot the L2 norm of the stress error vs DOFs. To this end, the stresses 39 are first computed, and then recovered using SPR theory. The von Mises stress is adopted 40 for this comparison. Again, the proposed HSchur decomposition outperforms the standard 41 eigen-decomposition scheme. However, certain features cannot be overcome. A change in 42 convergence rate is observed, when few element are used to model a side of the domain. 43 As stress recovery techniques require at least two elements to form a recovery patch, they 44 are not applicable when one element is used to discretize a side. The lack of stress recovery 45 options leads to a decreased rate of convergence. It is therefore proposed to discretize each 46 side by at least two elements, in order to harness the benefits of SPR theory. Furthermore, a 47 coarse discretization significantly limits the resolution of the mode shapes comprising part 48 of the SBFEM solution. 49 Since the stress variation in the tangential direction is not smooth around corners, special 50 consideration must be taken for these cases. In order to avoid errors, an automated procedure 51 is proposed, based on the orientation of adjacent elements. As long as the normal vectors of 52 two adjacent elements point along the same direction, i.e., the cosine of the angle spanned 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 29 4 L2 stress error, Eigen decomposition 5 10-1 6

7 10-2 8 10-3

9 r

r r 10 e s -4

s 10

e r

11 t

s 2 12 L 10-5 13 2-Nodes m = 1.63 3-Nodes m = 3.02

-6 4-Nodes m = 3.59 14 10 5-Nodes m = 4.66 15 6-Nodes m = 5.41 16 10-7 102 17 #DOF 18 Fig. 23 Convergence results of the L stress error based on the eigen-decomposition. 19 2 20 For Peer Review 21 L2 stress error, HSchur decomposition 22 10-1 23 24 10-2 25

10-3 r

26 o

r e

s -4

27 s 10

r t

28 s

2 L 29 10-5 2-Nodes m = 1.81 30 3-Nodes m = 3.26

-6 4-Nodes m = 3.73 10 31 5-Nodes m = 4.78 32 6-Nodes m = 5.64 10-7 33 102 34 #DOF

35 Fig. 24 Convergence results of the L2 stress error based on the HSchur decomposition. 36 37 38 39 between them approaches the value of 1, then the adjacent element is added to the patch. 40 Such a process may not be put in place for the case of corner points, an issue, which is 41 elaborated upon in the fourth numerical example. 42 Fig. 25 contrasts the von Mises stresses computed with only few DOF by SBFEM with 43 the exact stresses obtained from Table 3. The largest discrepancy on the domain results 44 around the four corners for the reason hinted at earlier. Hence, it it shown that optimal 45 superconvergence cannot be obtained due to this limitation. Apart from this exception, the 46 HSchur decomposition succeeds in improving the convergence behavior. 47 For the current numerical example, the stresses along the boundary in the direction of 48 crack extension are recoverable and thus the SIFs are accurately and efﬁciently calculated, 49 as observed in Fig. 26. The error is determined based on the exact values for the SIFs, im- 50 posed originally through the boundary conditions. The improvement over the case with only 51 trivial averaging of stresses at nodes (Fig. 27) is recognizable. The convergence rates m are 52 observed as superconvergent, i.e., the stresses converge at the same rate as the displacements. 53 54 55 56 57 58 59 60

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1 2 3 30 Adrian Egger et al. 4 5 6 7 8 9 10 11 12 13 14 15 16 Fig. 25 Qualitative location of errors in von Mises stress, by comparing the SBFEM solution to the analytical 17 ones provided in Table 3. Discretized with 3-noded elements and 4 elements per side. 18 19

Error in K2, recovered stress 20 1For Peer Review 10 21

22 100 23

24 10-1

2 K

25 n i -2

r 10

o r

26 r e

27 % 10-3 28 2-Nodes m = 2.04 3-Nodes m = 3.95

-4 4-Nodes m = 3.88 29 10 5-Nodes m = 4.84 30 6-Nodes m = 5.86

31 10-5 102 32 #DOF 33 K 34 Fig. 26 Percent error in 2 calculated using recovered stress. 35 36 37 38 Since one of the aims is to determine the potential of SBFEM for readily determining 39 the SIFs, the error in K2 is plotted against wall clock time for each analysis (Fig. 28). It 40 is apparent, that SBFEM can compute the SIFs with sub-percent accuracy in well under 41 a second on current generation computers. Further, it seems that the odd-noded elements 42 outperform their even-noded counter parts. 43 Table 5 summarizes the computational time required for completing the various tasks 44 in SBFEM. Striking is the fact that for the current implementation of SBFEM, recovery of 45 the stresses over the entire domain signiﬁcantly extends the required computational time. In 46 contrast, calculation of SIFs by the proposed local stress recovery scheme does not add any 47 signiﬁcant computational toll. 48 The associated error estimator for the SIFs is plotted in Fig. 29. Moreover, for this 49 example, the a posteriori error estimator is highly effective and may be easily determined. 50 In contrast to other SIF error estimators ([39],[40]), solving for a second right hand side is 51 not necessary. The absolute difference between estimated and exact error estimator is seen 52 to decrease for all elements and is generally in the order of percent fractions (Fig. 29). 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 31 4 Error in K2, averaged stress 5 101 6

7 100 8 10-1

9 2

K n

10 i -2

r 10

o r

11 r e

12 % 10-3 13 2-Nodes m = 2.04 3-Nodes m = 1.87

-4 4-Nodes m = 3.85 14 10 5-Nodes m = 4.38 15 6-Nodes m = 5.45 16 10-5 102 17 #DOF 18 Fig. 27 Percent error in K calculated using averaged stress. 19 2 20 For Peer Review 21 Error in K2, recovered stress 10 1

22 2-Nodes 3-Nodes 23 10 0 4-Nodes 5-Nodes 24 6-Nodes

25 10 -1 2

26 K

n i -2

r 10 o

27 r

r e

28 % 29 10 -3 30 10 -4 31

32 10 -5 10 -2 10 -1 10 0 10 1 10 2 33 seconds 34 35 Fig. 28 Percent error in K2 calculated using averaged stresses vs wall clock time. The calculations wer performed on an Intel Xeon E3-1225 v3. 36 37 38 Table 5 Percentage of computational time spent on various tasks for the solution of SBFEM. Discretized with 39 3-noded elements and 3 elements perside, 4-nodd elements and 4 elements per side and 5-noded elements 40 and 5 elements per side respectively. 41 Task 98 DOF 198 DOF 322 DOF 42 invert [E0] 0.53 0.63 0.43 43 HSchur 52.26 64.03 70.18 form K 0.38 0.30 0.25 44 invert K 6.87 3.88 2.64 45 form C 3.69 0.43 0.37 46 stress recovery 36.03 30.63 26.11 47 SIF 0.24 0.10 0.02 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 32 Adrian Egger et al. 4 Error estimator for K2 5 10 2 6 7 10 1

8 10 0

9 2

K 10 -1 n

10 i

r o

r 2-Nodes ext 11 r -2 e 10 3-Nodes ext

% 4-Nodes ext 12 5-Nodes ext 10 -3 6-Nodes ext 13 2-Nodes est 3-Nodes est 14 4-Nodes est -4 10 5-Nodes est 15 6-Nodes est 16 10 -5 10 2 17 #DOF 18 Fig. 29 Error estimator for K based on difference between raw and recovered stress. 19 2 20 For Peer Review 21 σ = 1 22 23 24 25 L a = L 26 4 E = 200 [N/mm2] 27 ν = 0.3 28

29 L 30 Fig. 30 Double edge cracked plate under tension. Problem statement for the third numerical example. 31 32 33 34 3.3 Double edge-cracked plate under tension 35 36 A double edge-cracked plate under tension as depicted in Fig. 30 is considered, with length 37 L = 1 units and crack length a = L/4. The boundary is uniformly discretized. For this numer- 38 ical example, SBFEM is compared to XFEM and the commercial FEM software Abaqus. 39 Three primary metrics for a solution, i.e., accuracy of calculated SIFs, floating point op- 40 erations (flops) and number of DOFs are considered. Since the motivation for employing 41 SBFEM to model fracture is accelerating simulations, analysis time is included as a sec- 42 ondary metric. 43 The flops required for the solution of each method are based on simplified assumptions. 44 Steps exist, such as the inversion of matrices and the solution of the eigen-problem, which 45 dominate the computational toll of an analysis. For XFEM and the standard FEM, this toll 46 is assumed to stem primarily from the inversion of the structures stiffness matrix. In SB- 47 FEM however, multiple computationally intensive steps exist, summarized in table 6. The 48 variables n and m denote the DOFs present in SBFEM and XFEM respectively, whereas b 49 represents the half-bandwidth of the matrix [E0] (Eqn. 15). The solution process in SBFEM 50 is constrained by the computational demands of the Schur decomposition. The newly pro- 51 posed HSchur decomposition reduces this burden by exploiting the symmetry conditions of 52 the Hamiltonian matrix [Z]. Considering flops alone, the HSchur decomposition is clearly 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 33 4 5 Table 6 Rough flop counts for two SBFEM variants and (X)FEM 6 SBFEM Eqn. # (X)FEM 7 Schur HSchur 8 23 bn2 bn2 0 3 3 2 9 22 25 2n 40n + 205n 0 24 3∗n3 3n3 0 10 diagonalize 2n3 2n3 0 11 29 2/3n3 2/3n3 0 1 3 3 3 12 U = K− F 1/3n 1/3n 1/3m 13 27 1/3n3 1/3n3 0 Total 619 n3 bn2 139 n3 205 b n2 1 3 m3 14 3 + 3 + ( + ) ( / ) 15 16 17 more performant. Since the method is currently implemented as proof-of-concept code writ- 18 ten in Matlab, execution times are similar to the generic Schur decomposition. Once this 19 novel method is assimilated into high performance libraries, calculation times are expected 20 to improve considerably. For Peer Review 21 In Abaqus, quadratic elements are used with the mid-side nodes set at the quarter points 22 and the element sides are collapsed into a single node, in order to emulate singular elements 23 capable of representing a square root singularity. The SIFs are calculated by a contour inte- 24 gral with four contours. In XFEM, linear elements are employed. The radius of enrichment 25 is kept constant at re = 0.2 units, while the radius of integration is fixed at ri = 2.55 times 26 the element size. The SIFs are calculated by interaction integral. In SBFEM, the use of 27 higher order elements does not significantly alter the computational effort. Since the stiff- 28 ness matrix is fully populated, higher order elements do not affect the bandwidth other than 29 0 1 in the formulation of [E ]− (Eqn. 23). This operation constitutes a minor part of the over- 30 all computational complexity (Table 6). Subsequently, the worst case is assumed and the 31 half-bandwidth is chosen to be maximal. The computational burden due to fully populated 32 matrices is offset by symmetry and the fact that only the boundary need be discretized. 33 Fig. 31 depicts the theoretical flop count required for analysis as a function of DOFs. 34 A comparison focused simply on flop count favors the XFEM and FEM-based implemen- 35 tations. The proposed HSchur decomposition performs favorably, compared to the current 36 standard Schur, requiring roughly an order of magnitude less flops for the same solution. 37 However, in a direct comparison to XFEM, it still requires substantially higher computa- 38 tional effort at an equal level of discretization. 39 In order to offer a more meaningful comparison, Fig. 32 provides the corresponding 40 accuracy of the calculated SIFs plotted against the DOFs. To this end, a reference solution 41 obtained from a fine mesh in Abaqus of approximately half a million DOFs is used. SB- 42 FEM evidently outperforms the competing methods, as fewer DOFs are required in order 43 to approach the reference solution. Significantly higher rates of convergence are obtained 44 by SBFEM by employing higher order elements. Noteworthy are the linear, two noded el- 45 ements, which outperform XFEM and the standard FEM approach with a slope of m = 2, 46 calculated using recovered stresses. 47 Fig. 33 depicts the error in calculated SIF as a function of the flops required for analy- 48 sis. At lower flop counts, all methods perform comparably. A slight edge goes to the FEM 49 implementation. Since in Abaqus the integration domain is specified manually, the amount 50 of DOFs present in the system is not uniformly distributed at coarser discretizations. Conse- 51 quently, the FEM plot exhibits a plateau. Once this plateau is passed, the rate of convergence 52 stabilizes to mFEM = 0.88. The rate of convergence for XFEM is higher at mXFEM = 1.06. 53 54 55 56 57 58 59 60

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1 2 3 34 Adrian Egger et al. 4 Computational requirements for solution 5 10 18 6 17 7 10 8 10 16

9 10 15

10 s

p 14

o 10 11 . 12 10 13

13 10 12

14 (X)FEM 10 11 SBFEM Schur 15 SBFEM HSchur 16 10 10 0 5 10 15 17 #DOF #10 4 18 Fig. 31 Computational requirements for solution for (X)FEM, SBFEM using the conventional Schur decom- 19 position and the newly proposed HSchur decomposition. 20 For Peer Review 21

22 Error in K1 vs #DOF 23 2-Nodes 24 100 3-Nodes 4-Nodes 25 5-Nodes 6-Nodes 26 FEM

1 XFEM

27 K 10-1

i r

28 o

r e

29 %

30 10-2 31 32

33 10-3 102 103 104 105 34 #DOF 35 36 Fig. 32 Accuracy of K1 as a function of DOFs for SBFEM, XFEM and FEM for the double edge crack 37 problem. For SBFEM the proposed HSchur decomposition is utilized. 38 39 40 At higher flop counts, SBFEM begins to separate itself from the competing methods, due to 41 higher rates of convergence. 42 Fig. 34 indicates the error in computed SIFs as a function of wall clock time. For Abaqus, 43 analysis times are extracted from the log file. For XFEM and SBFEM, the stopwatch timer 44 in Matlab is employed. Results for this numerical example are obtained on an Intel i5- 45 6600K @ 3.9 GHz. As observed in Fig. 34, the current implementation of SBFEM, based 46 on the newly adopted HSchur, outperforms XFEM by approximately an order of magnitude. 47 This quantification must be treated with caution, since both the SBFEM and XFEM imple- 48 mentations have yet to undergo optimization. This discrepancy between analysis time and 49 computation complexity (Fig. 33) is attributed to operations specific to XFEM, not incorpo- 50 rated in the theoretical flop count, and language related performance limitations. Examples 51 for the former are operations associated with the integration of the enrichment terms and 52 the post-processing of the SIFs. Employing loops in scripting languages, such as Matlab, is 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 35 4 Error in K1 vs .ops 5 10 1

6 2-Nodes 3-Nodes 7 4-Nodes 5-Nodes 8 10 0 6-Nodes FEM XFEM

9 1

K n

10 i -1

r 10

o r

11 r e

12 %

13 10 -2 14 15 16 10 -3 10 6 10 8 10 10 10 12 17 flops 18 Fig. 33 Accuracy of K1 as a function of the ﬂops required to obtain the solution for the double edge crack 19 problem. For SBFEM the proposed HSchur decomposition is utilized. 20 For Peer Review 21

22 Error in K1 vs time 1 23 10 24

25 10 0

26 1

27 K

n i -1

r 10 o

28 r

r e

29 % 2-Nodes 30 3-Nodes 10 -2 4-Nodes 31 5-Nodes 6-Nodes 32 FEM XFEM

33 10 -3 10 -2 10 -1 10 0 10 1 10 2 34 seconds 35 36 Fig. 34 Accuracy of K1 as a function of analysis wall clock time for the double edge crack problem. For SBFEM the proposed HSchur decomposition is utilized. 37 38 39 40 known to significantly impact run time. The number of loops performed during analysis is 41 dependant on the amount of elements and integration points, which is significantly larger for 42 the case of XFEM than SBFEM. These considerations are equally applicable to the fourth 43 numerical example. 44 Compared to the optimized code of the commercial software Abaqus, which utilizes 45 highly optimized numerical routines, SBFEM compares favorably. Similar accuracy in the 46 calculated SIFs is observed at comparable analysis times. Further, for the higher noded ele- 47 ments, SBFEM with the adopted HSchur decomposition manages to outperform commercial 48 software. 49 Fig. 35 compares the accuracy of the calculated SIFs to the computational effort by 50 proxy of flop count. The computational advantage of the HSchur decomposition over the 51 generic Schur decomposition is demonstrated to approach an order of magnitude, with the 52 analysis time expected to decrease with further refinement of the code. 53 54 55 56 57 58 59 60

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1 2 3 36 Adrian Egger et al. 4 Computational advantage of HSchur 5 10 1 6 7 8 10 0

9 1

K n

10 i -1

r 10 o

r 2-N HSchur 11 r e 3-N HSchur

% 4-N HSchur 12 5-N HSchur 6-N HSchur 13 10 -2 2-N Schur 3-N Schur 14 4-N Schur 5-N Schur 15 6-N Schur 16 10 -3 10 7 10 8 10 9 10 10 17 flops 18 Fig. 35 Accuracy of K1 comparing the proposed HSchur and standard Schur decomposition, based solely on 19 theoretical ﬂop counts. 20 For Peer Review 21 Comparison of condition number evolution 22 1012 23 24 1010 25 26

108 )

27 K

(

d n

28 o c 29 106 2-N HSchur 30 3-N HSchur 4-N HSchur 4 31 10 5-N HSchur 6-N HSchur 32 XFEM 33 102 101 102 103 104 34 # DOF 35 Fig. 36 Evolution of the condition number of the stiffness matrix as a function of DOFs for the double edge 36 crack problem. 37 38 39 A last comparison between XFEM and SBFEM considers the evolution of the condition 40 number of the stiffness matrix for an increasing number of DOFs. Fig. 36 demonstrates 41 the numerical difﬁculties associated with standard XFEM. Typically, the condition number 42 of the stiffness matrix increases rapidly, mandating preconditioning, which constitutes a 43 computational burden. This is not the case in SBFEM since the method naturally extends to 44 encompass fracture related phenomena. Consequently, no terms are added, which affect the 45 condition number adversely. However, it is evident that higher order elements do increase 46 the condition number steadily. 47 48 49 3.4 Slant crack in square plate 50 51 A slant crack as depicted in Fig. 37 is considered, with length L = 1 unit, crack length a = 52 L/2 and crack inclination angle equal to π/4. The boundary is evenly discretized. A crack 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 37 4 σ = 1 5 6

4 7 L/ = π/4 8 a L 9

10 E = 200 [N/mm2] 11 ν = 0.3 12 13 L 14 Fig. 37 Slant crack in square plate under tension. The problem statement for the fourth numerical example. 15 16 17 18 19 20 For Peer Review 21 22 23 24 25 Fig. 38 Sample discretization scheme for modeling a slant crack with SBFEM. It can be seen, that in order 26 to accommodate different crack inclination angles, simply the scaling centers (circles) must be moved, which 27 may be done independently of the mesh (crosses). 28 29 30 inclination angle of π/4 is purposely chosen, since the largest errors in computed stresses 31 appear at corner points (Fig. 25). The aim of this numerical example is to ascertain, how 32 SBFEM performs in a worst case scenario, when compared against alternatives. XFEM and 33 the standard FE-based approach are considered in the following comparisons. For XFEM, 34 the enrichment radius is kept constant at re = 0.2 units, while the radius of integration is 35 fixed at ri = 2.55 times the element size. For Abaqus, the same quadratic elements where 36 chosen as in the previous example. Modeling the slant crack problem with SBFEM entails 37 separating the domain into two parts. Each is inscribed its own scaling center, placed at the 38 crack tip (Fig. 38). The results reported, correspond to the SIFs calculated at the upper right 39 sided crack tip. 40 Fig. 39 illustrates the accuracy of computed K2 as a function of DOFs. To this end, a 41 finely meshed reference solution in Abaqus was employed. The FEM-based calculation of 42 the SIFs by Abaqus, the XFEM implementation as well as the 2-noded elements of SBFEM 43 are observed to converge at a comparable rate, although XFEM has a slightly higher rate 44 of convergence. Admitting higher order elements, SBFEM outperforms the other numeri- 45 cal methods. The odd-noded elements outperform their even-noded counterparts. Since the 46 computational effort per DOF for SBFEM is greater than for the two contrasted numerical 47 methods, Fig. 40 plots the accuracy of K2 as a function of flops. SBFEM compares favorably 48 to the FE-based approach and the XFEM implementation, even in this worst-case scenario. 49 It would seem that the calculation of SIFs at a crack inclination angle of π/4 is also a chal- 50 lenging task for other FE-based numerical methods. 51 Fig. 41 provides the analysis times for calculating the SIFs based on all three compared 52 methods. Calculations were performed on an Intel Xeon E3-1225 v3. Oscillatory behavior 53 54 55 56 57 58 59 60

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1 2 3 38 Adrian Egger et al. 4 Error in K2 vs #DOF 5 10 1 6 7 8 10 0

9 2

K n

10 i -1

r 10

o r

11 r e

12 % 2-Nodes 3-Nodes 4-Nodes 13 10 -2 5-Nodes 6-Nodes 14 FEM 15 XFEM 16 10 -3 10 1 10 2 10 3 10 4 10 5 10 6 17 #DOF 18 Fig. 39 Accuracy of K2 as a function of DOFs for the slant crack problem. For SBFEM the proposed HSchur 19 decomposition is utilized. 20 For Peer Review 21 22 23 for the first few points is attributed to the method of reporting in Abaqus. Values are reported 24 at discrete 0.1 second intervals. Since for systems with small amounts of DOFs the I/O 25 overhead dominates the overall computational time, oscillatory values of 0.2 and 0.3 seconds 26 were obtained. SBFEM is demonstrated to perform on par with current commercial software 27 in terms of efficiently and accurately calculating the SIFs. Further, for problems with simple 28 crack geometries, it outperforms a similarly implemented XFEM by approximately an order 29 of magnitude at equivalent accuracy. This is attributed to the ease with which higher order 30 elements may be employed. However, considering only linear elements, the advantage of 31 SBFEM over XFEM is marginal. 32 Fig. 42 plots K2 calculated based on raw stresses and the standard eigen-decomposition 33 as a function of analysis time. Especially the lower noded elements suffer from this transi- 34 tion. While the XFEM implementation outperforms the 2-noded elements, the commercial 35 software Abaqus performs similarly to the 3-noded elements. SBFEM is only observed to 36 maintain its competitive advantage, when employing elements with four or more nodes. 37 In Fig. 43 the condition number of an SBFEM solution is contrasted to that of XFEM 38 as a function of increasing DOFs. The condition numbers are not clearly separated. For 39 finer domain discretizations, the condition number of SBFEM approaches that of XFEM. 40 However, at that level of discretization, SBFEM is fully converged as opposed to XFEM. 41 This escalation in condition number is dependent on the angle spanned between an element 42 on the boundary and the scaling center. The smaller the angle, the more numerical difficulties 43 occur. With a higher boundary discretization, these angles tend to diminish. Should severe 44 numerical issues be encountered, a splitting of the domain perpendicular to the crack line 45 is advised, distributing the angles spanned by the boundary elements and the scaling center 46 more evenly. 47 One of the strengths of SBFEM is exploited to perform a study of variants. Modeling 48 of varying crack inclination angles becomes trivial, as only the scaling centers must be 49 repositioned. As a result, a procedure is proposed, which accommodates the application of 50 SPR theory to the SIFs. 51 52 1. The direction of crack extension is determined. 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 39 4 Error in K2 vs .ops 5 10 1 6 7 8 10 0

9 2

K n

10 i -1

r 10

o r

11 r e

12 % 2-Nodes 3-Nodes 4-Nodes 13 10 -2 5-Nodes 6-Nodes 14 FEM 15 XFEM 16 10 -3 10 6 10 8 10 10 10 12 17 flops 18 Fig. 40 Accuracy of K2 as a function of ﬂops required for solution of the slant crack problem. For SBFEM 19 the proposed HSchur decomposition is utilized. 20 For Peer Review 21

Error in K2 vs time 22 1 10 23 24

25 10 0

2 K

27 n i -1

r 10

o r

28 r e

29 % 2-Nodes 30 3-Nodes 10 -2 4-Nodes 5-Nodes 31 6-Nodes FEM 32 XFEM

33 10 -3 10 -2 10 -1 10 0 10 1 10 2 34 seconds 35 Fig. 41 Accuracy of K2 as a function of analysis wall clock time for the slant crack problem. For SBFEM 36 the proposed HSchur decomposition is utilized. 37 38 39 40 2. The intersection point between the boundary and the crack extension line is calculated. 41 The nearest node is calculated and the corresponding intersection element is found. 42 3. The stresses are recovered over a patch of two elements, as described in Sec. 2.3, with 43 the addition of a check, ensuring the chosen patch does not entail corners. This process 44 is automated by applying the orientation checking procedure as detailed in Sec. 3.2. 45 4. The SIFs are calculated based on the locally recovered stresses, which are rotated by the 46 crack inclination angle.

47 In Table 7 the SIF K1 is tabulated for various discretization levels and crack inclination 48 angles. A reference solution is obtained from a ﬁne meshed Abaqus analysis. In order to 49 investigate the stability of SBFEM for various crack inclination angles, the values for K1 50 are normalized with respect to the reference solution. As observed in Table 7, K1 behaves 51 independently of the crack inclination angle. All three methods, SBFEM, XFEM and FEM 52 provide comparable results, with deviations lower than 1%. 53 54 55 56 57 58 59 60

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1 2 3 40 Adrian Egger et al. 4 Error in K2 vs time 5 10 1 6 7 8 10 0

9 2

K n

10 i -1

r 10

o r

11 r e

12 % 2-Nodes 3-Nodes 4-Nodes 13 10 -2 5-Nodes 6-Nodes 14 FEM 15 XFEM 16 10 -3 10 -2 10 -1 10 0 10 1 10 2 17 seconds 18 Fig. 42 Accuracy of K2 as a function of analysis wall clock time for the slant crack problem. For SBFEM 19 the eigen-decomposition combined with raw stresses for the calculation of K2 is utilized. 20 For Peer Review 21 Comparison of condition number evolution 22 10 12 23 24 10 10 25 26

10 8 )

27 K

(

d n

28 o c 29 10 6 2-N HSchur 30 3-N HSchur 10 4 4-N HSchur 31 5-N HSchur 6-N HSchur 32 XFEM

33 10 2 10 2 10 3 10 4 34 # DOF 35 Fig. 43 Evolution of the condition number of the stiffness matrix as a function of DOFs for the slant crack 36 problem. 37 38 39 Table 7 Values of computed K1 as a function of the crack inclination angle ranging from 10◦ to 80◦. For 40 SBFEM, 3-noded elements were chosen with 1 to 4 elements discretizing each side. For Abaqus and XFEM, K1 was calculated using ﬁne meshes. The normed values correspond to the ratio of SBFEM K1 solution to 41 the Abaqus reference solution. 42 43 44 Type 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 45 36 DOFs 1.09394 1.03033 0.87814 0.69875 0.50031 0.31072 0.15275 0.05671 46 50 DOFs 1.09307 1.02125 0.87838 0.69869 0.50038 0.31028 0.15225 0.04827 47 74 DOFs 1.09390 1.01546 0.87856 0.69873 0.50042 0.31006 0.15200 0.04691 98 DOFs 1.09486 1.01544 0.87871 0.69865 0.50046 0.31001 0.15189 0.04682 48 Normed 1.00475 1.00444 1.00581 1.00735 1.00538 1.00515 0.99140 0.99272 49 Abaqus ref. 1.08968 1.01094 0.87364 0.69355 0.49778 0.30843 0.15321 0.04716 50 XFEM 1.09278 1.01336 0.87647 0.69119 0.50215 0.30976 0.15172 0.04707 51 52 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 41 4 5 Table 8 Values of computed K2 as a function of the crack inclination angle ranging from 10◦ to 80◦ using 6 the same methodology as in Table 7. 7 8 Type 10◦ 20◦ 30◦ 40◦ 50◦ 60◦ 70◦ 80◦ 9 36 DOFs 0.13061 0.29445 0.41963 0.49134 0.50066 0.44532 0.33312 0.17540 10 50 DOFs 0.13018 0.29185 0.41719 0.49132 0.50072 0.44541 0.33329 0.17823 11 74 DOFs 0.12990 0.29034 0.41713 0.49123 0.50073 0.44569 0.33344 0.17862 12 98 DOFs 0.12935 0.28989 0.41703 0.49126 0.50073 0.44572 0.33348 0.17862 13 Normed 0.99768 1.00279 1.00322 1.00338 1.00190 1.00051 0.99779 0.99700 14 Abaqus ref. 0.12965 0.28908 0.41569 0.48961 0.49978 0.44550 0.33422 0.17915 XFEM 0.12946 0.28948 0.41683 0.49037 0.49991 0.44586 0.33454 0.17923 15 16 17 18 Table 8 summarizes the results for the case of K2. It is verified that the SIFs computed 19 by SBFEM are independent of the crack inclination angle. However, as the crack inclination 20 angle is increased, numericalFor issues arise duePeer to the introduction Review of small angles spanned by 21 the elements on the boundary and the scaling center. These angles approach the values of 22 zero as the crack inclination angle approaches 90◦ (Fig. 38). 23 24 25 4 Conclusion 26 27 In this work, the adoption of a blocked Hamiltonian Schur decomposition is proposed, which 28 proves highly beneficial when substituted into the solution process of SBFEM. By preserv- 29 ing Hamiltonian symmetry, i.e., the symmetry of the eigenvalues about the real and the 30 imaginary axis, enhanced accuracy is achieved leading in improved, near optimal, conver- 31 gence rates. Further, only half of the analysis domain need be considered, which alleviates 32 the additional computational toll incurred in linearizing the quadratic eigenvalue problem. 33 An additional benefit stems from the fact that the eigenvalues are automatically partitioned 34 based on sign. Thus, errors incurred due to improper sorting, resulting in an erroneous stiff- 35 ness matrix are eliminated along with the prior need for a stabilizing parameter ε. By con- 36 sequence, the adopted HSchur decomposition is demonstrated to significantly improve the 37 regularity of the method. 38 The convergence rates of SBFEM are investigated in a benchmark example, featuring an 39 analytical solution. The L2 error in displacements, stresses as well as the error in computed 40 stress intensity factors are recovered. Implementation of SPR theory allows for supercon- 41 vergent recovery of stresses, and in select cases ultracovergent behavior. This property is 42 exploited to significantly improve the accuracy and convergence behavior of the calculated 43 SIFs. It is further demonstrated that stress recovery need not be carried out for the entire 44 domain, since similar accuracy is achieved when only considering the patch in direction of 45 crack extension, at reduced computational cost. A local error estimator is proposed, which 46 arises as a natural extension of the stress recovery scheme, and is highly accurate as well as 47 computationally inexpensive. 48 Finally, SBFEM is explicitly contrasted against other numerical methods within the con- 49 text of LEFM in terms of rough flop count, and wall clock time (albeit for unoptimized Mat- 50 lab implementations for the SBFEM and XFEM variants). It is demonstrated that for simple 51 crack geometries, a higher order SBFEM consistently outperforms frequently adopted alter- 52 natives. 53 54 55 56 57 58 59 60

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1 2 3 42 Adrian Egger et al. 4 5 Acknowledgements This research was performed under the auspices of the Swiss National Science Founda- 6 tion (SNSF), Grant # 200021 153379, A Multiscale Hysteretic XFEM Scheme for the Analysis of Composite Structures. 7 The authors extend their gratitude to Dr. Konstantinos Agathos, of the University of Luxembourg, for 8 providing the Matlab R source code of the XFEM implementation as well as Prof. Dr. Volker Mehrmann 9 from TU Berlin and Prof. Dr. David Watkins, of Washington State University, for providing the Hamiltonian 10 Schur algorithm. 11 12 Appendix A.1 13 14 Considering the term representing the internal virtual work ﬁrst, it is rearranged after substituting Eqns. 9 and 15 11: 16 Z δε(ξ,η) T σ(ξ,η) dV 17 V { } { } Z T 18 1 1 2 = [B (η)] δu(ξ) ,ξ + [B (η)] δu(ξ) 19 V { } ξ { } 20 For Peer1 Review [D] [B1(η)] u(ξ) + [B2(η)] u(ξ) dV 21 × { },ξ ξ { } Z Z ξ=1 22 T 1 T 1 = δu(ξ) ,ξ [B (η)] [D][B (η)]ξ u(ξ) ,ξ J dξdη 23 ∂Ω ξ=0 { } { } | | 24 Z Z ξ=1 T 1 T 2 + δu(ξ) ,ξ [B (η)] [D][B (η)] u(ξ) J dξdη 25 ∂Ω ξ=0 { } { }| | 26 Z Z ξ=1 T 2 T 1 + δu(ξ) [B (η)] [D][B (η)] u(ξ) ,ξ J dξdη 27 ∂Ω ξ=0 { } { } | | 28 Z Z ξ=1 1 + δu(ξ) T [B2(η)]T [D][B2(η)] u(ξ) J dξdη (67) 29 ∂Ω ξ=0 { } ξ { }| | 30 Green’s theorem leads to the following formulation: 31 Z 32 δε(ξ,η) T σ(ξ,η) dV 33 V { } { } Z T 1 T 1 34 = δu(ξ) [B (η)] [D][B (η)]ξ u(ξ) ,ξ J dη ∂Ω { } { } | | ξ=1 35 Z 36 δu(ξ) T [B1(η)]T [D][B1(η)] − ∂Ω { } 37 u(ξ) + u(ξ) J dξdη × {{ }ξ { }ξξ }| | 38 Z + δu(ξ) T [B1(η)]T [D][B2(η)] u(ξ) J dη 39 ∂Ω { } { }| | ξ=1 40 Z Z ξ=1 T 1 T 2 δu(ξ) [B (η)] [D][B (η)] u(ξ) ,ξ J dξdη 41 − ∂Ω ξ=0 { } { } | | 42 Z Z ξ=1 T 2 T 1 + δu(ξ) [B (η)] [D][B (η)] u(ξ) ,ξ J dξdη 43 ∂Ω ξ=0 { } { } | | 44 Z Z ξ=1 1 45 + δu(ξ) T [B2(η)]T [D][B2(η)] u(ξ) J dξdη (68) ∂Ω ξ=0 { } ξ { }| | 46 In order to simplify the above equation, the following substitutions are introduced: 47 Z 48 [E0] = [B1(η)]T [D][B1(η)] J dη (69a) ∂Ω | | 49 Z 50 [E1] = [B1(η)]T [D][B2(η)] J dη (69b) ∂Ω | | 51 Z [E2] = [B2(η)]T [D][B2(η)] J dη (69c) 52 ∂Ω | | 53 54 55 56 57 58 59 60

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1 2 3 An Enhanced Scaled Boundary Finite Element Method for Linear Elastic Fracture 43 4 5 These simplifications are named “coefficient matrices” and resemble in structure the stiffness matrix of stan- 6 dard FEM schemes. They are calculated element wise and then assembled in the standard FEM sense. Ap- plying boundary conditions on the coefficient matrices is premature, as this will effectively delete parts of 7 the domain corresponding to the space spanned by the DOFs on the boundary and the scaling center. Using 8 the same notation to denote the fully assembled coefficient matrices, the previous equations are rewritten as follows using an abbreviation on the boundary u =ˆ u(ξ = 1) : 9 { } { } 10 Z 11 δε(ξ,η) T σ(ξ,η) dV V { } { } 12 = δu T [E0] u + [E1]T u 13 { } { { },ξ { }} Z ξ=1 T n 0 14 δu(ξ) [E ]ξ u(ξ) ,ξξ 15 − xi=0 { } { } 1 o + [[E0] + [E1]T [E1]] u(ξ) [E2] u(ξ) dξ (70) 16 − { },ξ − ξ { } 17 18 19 References 20 For Peer Review 21 1. K.J. Bathe, Finite element procedures (Bathe, Boston, Mass., 2006). OCLC: 732251900 2. T.J.R. Hughes, The finite element method: linear static and dynamic finite element analysis (Dover Pub- 22 lications, Mineola, NY, 2000) 23 3. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The finite element method: its basis and fundamentals, seventh 24 edition edn. (Elsevier, Butterworth-Heinemann, Amsterdam, 2013) 25 4. M. Kuna, Finite elements in fracture mechanics: theory - numerics - applications. No. 201 in Solid mechanics and its applications (Springer, Dordrecht, 2013). OCLC: 858004580 26 5. C. Brebbia, J. Dominguez, Applied Mathematical Modelling 1(7), 372 (1977). DOI 10.1016/ 27 0307-904X(77)90046-4 28 6. W.S. Ventu, Boundary element method in geomechanics. No. 4 in Lecture Notes in engineering (Springer, 29 Berlin, 1983) 7. P. Cundall, in Proc. Int. Symp. Rock Fracture, ISRM (Nancy (F), 1971), pp. 2–8 30 8. B.N. Rao, S. Rahman, Computational Mechanics 26(4), 398 (2000). DOI 10.1007/s004660000189 31 9. V.P. Nguyen, P. Kerfriden, S.P. Bordas, Composites Part B: Engineering 60, 193 (2014). DOI 10.1016/j. 32 compositesb.2013.12.018 10. M. Hofacker, C. Miehe, International Journal for Numerical Methods in Engineering 93(3), 276 (2013). 33 DOI 10.1002/nme.4387 34 11. N. Mos, J. Dolbow, T. Belytschko, International Journal for Numerical Methods in Engineering 46(1), 35 131 (1999). DOI 10.1002/(SICI)1097-0207(19990910)46:1 131::AID-NME726 3.0.CO;2-J h i 36 12. N. Sukumar, D. Chopp, N. Mos, T. Belytschko, Computer Methods in Applied Mechanics and Engineer- ing 190(46-47), 6183 (2001). DOI 10.1016/S0045-7825(01)00215-8 37 13. A. Gravouil, N. Mos, T. Belytschko, International Journal for Numerical Methods in Engineering 53(11), 38 2569 (2002). DOI 10.1002/nme.430 39 14. K. Agathos, E. Chatzi, S.P. Bordas, Computer Methods in Applied Mechanics and Engineering 306, 19 (2016). DOI 10.1016/j.cma.2016.03.023 40 15. V. Gupta, C. Duarte, I. Babuka, U. Banerjee, Computer Methods in Applied Mechanics and Engineering 41 266, 23 (2013). DOI 10.1016/j.cma.2013.07.010 42 16. K. Agathos, E. Chatzi, S.P.A. Bordas, D. Talaslidis, International Journal for Numerical Methods in 43 Engineering 105(9), 643 (2016). DOI 10.1002/nme.4982 17. S. Mohammadi, XFEM fracture analysis of composites (John Wiley & Sons, Ltd, Chichester, West Sus- 44 sex, United Kingdom, 2012) 45 18. P. Silvester, D. Lowther, C. Carpenter, E. Wyatt, Proceedings of the Institution of Electrical Engineers 46 124(12), 1267 (1977) 47 19. G. Dasgupta, Journal of Applied Mechanics 49(1), 136 (1982). DOI 10.1115/1.3161955 20. J.P. Wolf, C. Song, Computer Methods in Applied Mechanics and Engineering 123(1-4), 355 (1995). 48 DOI 10.1016/0045-7825(95)00781-U 49 21. J.P. Wolf, C. Song, Finite-element modelling of unbounded media (Wiley, Chichester, England ; New 50 York, 1996) 22. J.P. Wolf, The scaled boundary finite element method (J. Wiley, Chichester, West Sussex, England ; 51 Hoboken, NJ, USA, 2003) 52 23. A.J. Deeks, J.P. Wolf, Computational Mechanics 28(6), 489 (2002). DOI 10.1007/s00466-002-0314-2 53 54 55 56 57 58 59 60

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