A Trefftz collocation method (TCM) for three-dimensional linear elasticity by using the Papkovich-Neuber solutions with cylindrical harmonics Guannan Wang a, Leiting Dong b,*, Satya N. Atluri a a Center for Advanced Research in the Engineering Sciences, Texas Tech University, Lubbock, TX 79409, USA b School of Aeronautic Science and Engineering, Beihang University, Beijing, China Abstract A Trefftz collocation method (TCM) is proposed for solving three-dimensional (3D) linear-elastic boundary value problems. By using the Papkovich-Neuber (P-N) general solutions, Trefftz trial functions are expressed in terms of cylindrical harmonics. Both non-singular and singular harmonic functions are included, facilitating the study of interior and exterior problems. To mitigate the problem of ill-conditioning, two steps are adopted: The first step is to introduce a characteristic length of the domains of interests into the Laplace equation, and the second step is to scale each column of the coefficient matrix in the established system of linear equations using another multi-scale characteristic length, letting each column have the equal norms. Several examples are presented to validate the proposed 3D Trefftz collocation method. The completeness of the trial functions, the effect of the scaling techniques, and the accuracy of solutions are also discussed. Keywords: Trefftz method; 3D elasticity; Papkovich-Neuber solution; cylindrical harmonics; ill-conditioning 1. Introduction Trefftz method [1] is a meshless method for the numerical solutions of boundary value problems, where the trial functions are represented as a linear combination of functions that satisfy the governing differential equations a priori. Trefftz method can be categorized as direct and indirect formulations, depending on whether Trefftz basis functions are applied as test or trial functions [2]. The unknown coefficients in the Trefftz functions are usually determined by matching the boundary conditions. It has also been found that the collocation technique is the one of the simplest methods in dealing with boundary conditions and providing accurate numerical results [3-4]. Trefftz method has been successfully applied in numerous physical, mathematical and engineering situations, including Helmohotz equation [5-7], Poisson equation [8], Laplace equations [9-11], eigenvalue problems [7], elasticity problems [12], plate bending problems [13], free vibration problems [6]. Trefftz method was also associated with finite element (FE) in generating hybrid-Trefftz (HT) FE model since the early contributions by Jirousek and his coworkers [14-15]. Dong and Atluri applied Trefftz Voronoi Cell Finite elements in micromechanical modeling of heterogeneous materials with different shaped inclusions or voids [16-17], in which the complete trial functions predicted accurate effective properties and localized stress concentrations. A detailed review work of similar topics was conducted by Qin [18]. The 3D elasticity solutions have been studied for decades. To the best knowledge of the authors, the solutions proposed by [19-20] are the most complete solutions ever developed based on cylindrical coordinates until the present stage. However, these solutions only focused on the non-singular terms while ignoring the terms that cause singularity at the origins of the coordinates, which limits their usefullness for exterior problems. Based on the situation described above, along with the fact that the 3D elasticity P-N solutions using spherical harmonics and elliptical harmonics were developed by [16-17] for the purpose of solving spherical/elliptical shaped subject domains, we develop a complete 3D Trefftz method by employing P-N solutions that automatically satisfy Navier’s equations in terms of the cylindrical harmonics, including both non-singular and singular terms, in order to deal with not only interior but also exterior problems. The boundary conditions are applied through collocation techniques, where the P-N solutions are matched with the displacement or traction boundary conditions point by point. Despite of its wide applications, traditional Trefftz method still suffers from severe ill conditioning problems after establishing system of equations in certain scenarios. Liu [21-22] resolved the problem for Trefftz solutions of Laplace equation by introducing characteristic lengths and significantly reduced the condition numbers. Here similar procedures are adopted to rescale the cylindrical harmonic terms and the systems of equations, where two characteristic lengths are introduced. Finally, several elasticity examples within different domains of interests are presented to prove the accuracy and stability of the present method. The organization of the present work follows as: Section 2 exhibits the governing equations for 3D elasticity, and presents the P-N solutions in which Laplace potential is solved using cylindrical harmonics. Since unscaled functions of cylindrical harmonics include several types of functions, which will cause severely ill-conditioned results, two scaling techniques are introduced in Section 3 to alleviate the badly scaled conditions. Section 4 presents several numerical examples, proving the accuracy of Trefftz solutions and collocation technique, as well as the necessity of using scaling techniques. Section 5 concludes this presentation. 2. Trefftz trial functions for 3D linear elasticity 2.1 The Papkovich-Neuber solution The general solutions of 3D linear elasticity should satisfy the equations of stress equilibrium, strain-displacement gradient compatibility, as well as constitutive relations: ∇⋅σ +f =0 (1) 1 ∗ ε =[ ∇uu +∇ ( )] (2) 2 σ =λµtr()ε I + 2 ε (3) where σ , ε , and u are stresses, strains, and displacements, and T T T σ = [,,,,,]σσσσσσxx yy zz xy yz xz , ε = [,,,,,]εεεεεεxx yy zz xy yz xz and u = [,,]uuuxyz . ∇⋅ and ∇ refer to the divergence and gradient, respectively. f is the body force vector acting on elastic body; ()∇u ∗ is the transpose of ∇u ; I is the identity tensor; λ and µ are Lame’s constants given as λν=E (1 +− ν ) (1 2 ν ) and µν=E 2 (1 + ) for isotropic materials. Navier’s equations can be deduced from Eqs. (1-3), representing equilibrium equations in terms of displacement components [20,23]: (λµ+ )( ∇ ∇⋅u ) + µ ∇2 uf + = 0 (4) where ∇2 is the Laplacian operator. The difficulty of solving the 3D Navier’s equations arises from the coupled displacement components appearing in each of the equations in Eq. (4). Papkovich [24] and Neuber [25] indicated that the general solutions can be expressed in terms of harmonic functions if the body force is neglected: u=4(1 −v ) B −∇ ( RB ⋅ + B0 ) (5) in which B and B0 are vector and scalar harmonic functions (P-N potentials), and Re= xi i denotes the position vector. It can be easily noticed that the equilibrium equations, Eq. (4), only contain three unknown displacement components in Cartesian coordinate. Therefore, it is acceptable that only the vector harmonic function B (including three components) is kept, and dropping the scalar function B0 leads to a new solution: u=4(1 −v ) B −∇ RB ⋅ (6) which is a complete solution for the infinite domain. However, one can prove that the general solution of Eq. (4) can be solved as Eq. (6) only under the condition ν ≠ 0.25 for a simply-connected region. Later on M.G. Slobodyansky proved that for a simply connected domain, a complete general solution can be obtained by expressing B0 as a function of B [23], leading to: u=4(1 −v ) B + R ⋅∇ B −∇ RB ⋅ (7) 2.2 Cylindrical Harmonics as Papkovich-Neuber potentials Consider a position vector within Cartesian coordinates [,xxx123 , ] and the corresponding cylindrical coordinates [qqq123 , , ]= [,,] Rθ z, they have the relation: xR1=cosθθ , xR 23 = sin , xz = (8) The base vectors e12= eerz,, = eeθ 3 = e are in the directions of the radius of circles, the tangents to circles and the axis of the concentric cylinders, respectively. Now we have ∂∂xx ∂ x 11==−=cosθθ , R sin , 1 0 ∂∂Rzθ ∂ ∂∂xx ∂ x 22=sinθθ , = R cos , 2= 0 ∂∂Rzθ ∂ ∂∂∂xxx 333=0, = 0, = 1 ∂∂∂Rzθ (9) and ∂qs 1 ∂x ∂∂RR =k ⋅=δ 2 s, rsrsHH r s (10) ∂xks Hq ∂ ∂∂ q q where HH1 =r =1, HHr2 =θ = , HH3 =z =1 are referred to as Lame’s coefficients. The gradient of a scalar field can be expressed as: 1 ∂ψ ψ = = gradaasse , s s . (11) Hqs ∂ Then the Laplace equation in cylindrical coordinate can be obtained: 22 2 11∂∂ψ ∂ ψψ ∂ ∇=ψ r + + =0 (12) rr∂∂ r r22 ∂θ ∂ z 2 By separation of variables, the Laplace equation (12) can be decomposed into three differential equations with decoupled components, and different solutions are obtained corresponding to different values of parameters (see Appendix). Finally, the general solutions of the Laplace equation are in the non-singular form ψθin (,rz , )(finite at r = 0 for interior problems) and singular form ψθex (,rz , ) (infinite at r = 0 for exterior problems): ∞ ermmcos( mθθ )+ er sin( m ) ψθ(,rza , )=++ az 12mm in 01 02 ∑ mm m=1 ++e34mm zrcos( mθθ ) e zr sin( m ) ∞ c1nmcosh( nz ) J 02 ( nr )+ c sinh( nz ) J 0 ( nr ) +∑ n=1 ++c3mmcos( nz ) I 04 ( nr ) c sin( nz ) I 0 ( nr ) d12mn cos( mθ )cosh( nz ) Jm ( nr ) + d mn sin(mθ )cosh( nz ) Jm ( nr ) ∞∞ ++d34mn cos( mθθ )sinh( nz ) Jm ( nr ) dmn sin( m )sinh( nz ) Jm ( nr ) + ∑∑ ++θθ nm=11 = d56mn cos( m )cos( nz ) Im ( nr ) dmn sin( m )cos( nz ) Im ( nr ) ++d78mn cos( mθθ )sin( nz ) Im ( nr ) dmn sin( m )sin( nz ) Im ( nr ) (13) and −− ∞ ermmcos( mθθ )+ er sin( m ) ψθ(rza , , )=++ ln razr ln 56mm ex 03 04 ∑ −−mm m=1 ++e78mm zrcos( mθθ ) e zr sin( m ) ∞ c5nmcosh( nz ) Y 06 ( nr )+ c sinh( nz ) Y 0 ( nr ) +∑ n=1 ++c7mmcos( nz ) K 08 ( nr ) c sin( nz ) K 0 ( nr ) d9mn cos( mθ )cosh( nz ) Ym (nr )+ d10mn sin( mθ )cosh( nz ) Ym ( nr ) ∞∞ ++d11mn cos( mθθ )sinh( nz ) Ym ( nr ) d12mn sin( m )sinh( nz ) Ym ( nr ) + ∑∑ ++θθ nm=11 = d13mn cos( m )cos( nz ) Km ( nr ) d14mn sin( m )cos( nz ) Km ( nr ) ++d15mn cos( mθθ )sin( nz ) Km ( nr ) d16mn sin( m )sin( nz ) Km ( nr ) (14) Substituting the non-singular solution, Eq.