Topology Optimization for Designing Periodic Microstructures Based on ﬁnite Strain Viscoplasticity

Structural and Multidisciplinary Optimization (2020) 61:2501–2521 https://doi.org/10.1007/s00158-020-02555-x

RESEARCH PAPER

Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity

Niklas Ivarsson1 · Mathias Wallin1 · Daniel A. Tortorelli2

Received: 15 September 2019 / Revised: 20 February 2020 / Accepted: 21 February 2020 / Published online: 28 May 2020 © The Author(s) 2020

Abstract This paper presents a topology optimization framework for designing periodic viscoplastic microstructures under finite deformation. To demonstrate the framework, microstructures with tailored macroscopic mechanical properties, e.g., maximum viscoplastic energy absorption and prescribed zero contraction, are designed. The simulated macroscopic properties are obtained via homogenization wherein the unit cell constitutive model is based on finite strain isotropic hardening viscoplasticity. To solve the coupled equilibrium and constitutive equations, a nested Newton method is used together with an adaptive time-stepping scheme. A well-posed topology optimization problem is formulated by restriction using filtration which is implemented via a periodic version of the Helmholtz partial differential equation filter. The optimization problem is iteratively solved with the method of moving asymptotes, where the path-dependent sensitivities are derived using the adjoint method. The applicability of the framework is demonstrated by optimizing several two-dimensional continuum composites exposed to a wide range of macroscopic strains.

Keywords Topology optimization · Material design · Finite strain · Rate-dependent plasticity · Discrete adjoint sensitivity analysis

1 Introduction composite materials with enhanced properties, by optimizing the material microstructure. Examples of these include Topology optimization is a powerful tool that enables engi- linear elastic materials with negative Poisson’s ratio (Sig- neers to enhance structural performance. Since the work mund 1995), negative thermal expansion (Sigmund and by Bendsøe and Kikuchi N. (1988), topology optimization Torquato 1997), and extreme bulk modulus (Huang et al. has rapidly evolved and been applied to design a variety of 2011), and also viscoelastic materials with extreme bulk physical systems (Deaton and Grandhi 2014; Bendsoe and modulus (Andreasen et al. 2014). Sigmund 2013). Specifically, topological design methods In the design of such architected composite materials, based on the inverse homogenization approach by Sigmund a common simplification in the formulation is to use lin- (1994) have demonstrated the possibility to create novel ear elasticity, which implies that composite materials that exhibit irreversible processes and are exposed to large deformations cannot be designed. While linear assumptions Responsible Editor: YoonYoung Kim suffice for many composite design optimization applica- Niklas Ivarsson tions wherein the microstructure is subjected to moderate niklas.ivarsson@solid.lth.se macroscopic strain levels, they fail to accurately model the microstructural material response when it is subjected to Mathias Wallin larger macroscopic strains. To model this microstructural [email protected] response, nonlinear finite strain theory should be incorpo- Daniel A. Tortorelli rated into the composite topology optimization. [email protected] The evaluation of the homogenized material properties becomes demanding when accounting for response non- 1 Division of Solid Mechanics, Lund University, Lund, Sweden linearities, due to the need for iterative nonlinear analyses 2 Center for Design and Optimization, Lawrence Livermore (Yvonnet et al. 2009). Incorporating nonlinear simulations National Laboratory, Livermore, CA, USA into topology optimization problems is a formidable task. 2502 N. Ivarsson et al.

Even the efficient parallel programming computations, as into a topology optimization framework. Macroscopic non- presented in Nakshatrala et al. (2013), for the design of linear material properties are evaluated by numerical tensile nonlinear elastic composites, and the model reduction tech- and shear tests of a single unit cell that is subjected to peri- nique computations, as presented in Fritzen et al. (2016), for odic boundary conditions. Our framework builds upon the designing multiscale elastoviscoplastic structures, remain works of Ivarsson et al. (2018), as rate-dependent finite computationally intensive tasks. strain effects are included, but for simplicity, we neglect In the works by Wang et al. (2014) and Wallin and Tor- inertia. Contrary to our previous work, where structural torelli (2020), alternative nonlinear homogenization meth- design was considered, we here study microstructures opti- ods have been proposed to design architected materi- mized for maximum energy absorption. Thus, the present als. These homogenization methods use numerical tensile work allows each point in a macroscopic structure to be experiments for calculating the effective uniaxial material tailored to its specific loading rate, load path, and load response and use topology optimization to achieve desired magnitude it experiences. In particular, we investigate the macroscopic properties. In this way, the computational influence of nonproportional loading and distortion con- effort required for the analysis is significantly reduced, straints. Ultimately we quantify the optimization cost and since just a few desired properties are targeted. constraint functions via the response from a finite strain vis- Of the developed topology optimization methods, few coplastic boundary value problem which is solved using a consider structures exhibiting inelastic material response. nested Newton approach, wherein the momentum balance Contributions incorporating small strain elastoplastic for- equation is coupled with local constitutive equations. We mulations include the work by Maute et al. (1998), where discretize the kinematics using a total Lagrangian formu- topology optimization is used to optimize structural ductil- lation and use the backward Euler scheme to integrate the ity. Conceptual designs of energy absorbers for crashworthi- constitutive equations. ness, modeled with 2D-beam elements, were established by To formulate a well-posed topology optimization prob- Pedersen (2004). Protective systems with maximum energy lem, we use restriction wherein we constrain the feature dissipation for continuum structures subject to impact load- length scale in the volume fraction field by mollifying it ing have also been designed by Nakshatrala and Tortorelli via a periodic version of the Helmholtz partial differential (2015), where a transient elastoplastic topology optimiza- equation (PDE) filter (Lazarov and Sigmund 2011). The tion formulation is used. Elastoplastic material modeling optimization problem is solved by the method of moving has further been used in Bogomolny and Amir (2012) asymptotes (MMA), which requires gradients to form the for optimizing steel-reinforced concrete structures, in Kato convex MMA approximation. For path-dependent material et al. (2015) for optimizing composite structures for maxi- response, it is challenging to compute response sensitivities. mum energy absorption, and more recently in Wallin et al. This is because the sensitivities depend on the response his- (2016) for maximizing the energy absorbtion of structures tory, as opposed to, e.g., elasticity, where they do not. To under finite deformation. Also, in our previous paper (Ivars- compute the gradients, we use the adjoint sensitivity analy- son et al. 2018), we combine dynamic rate-dependent finite sis because the design variables in the optimization problem strain effects with topology optimization to design max- greatly outnumber the constraints. Several studies (Bogo- imal energy absorbing structures. In the design of such molny and Amir 2012; Wallin et al. 2016; Zhang et al. energy absorbing structures, e.g., bumpers in automotive 2017;Amir2017; Ivarsson et al. 2018; Zhang and Khan- engineering, it may also be desirable to control the volume delwal 2019) of topology optimization for path-dependent change, since devices that operate in confined spaces will problems have successfully adopted the adjoint sensitivity be extremely stiff if the deformation is associated with a recipe presented in Michaleris et al. (1994). This requires volume increase. the evaluation and storage of the primal response trajec- To date, most topology optimization frameworks that tory, and the solution of a terminal-valued adjoint sensitivity incorporate nonlinearities focus on macroscopic design; problem. We implement this adjoint method to compute the research on composite material design via inverse homog- sensitivities. enization that incorporates inelastic constituents is scarce. The remainder of this paper is organized as follows: Examples of such studies include the recent works by Chen In Section 2, we present the kinematics and governing et al. (2018) and Alberdi and Khandelwal (2019)which mechanical balance equations. Section 3 summarizes the assume infinitesimal deformation. In this paper, we generate constitutive model and Section 4 presents the numerical optimal topologies of periodic microstructures for maxi- solution procedure and the periodic boundary condition mum energy dissipation with possible prescribed zero con- implementation. Section 5 describes the regularization and traction constraints under finite deformation. To do this, we material interpolation schemes. This is followed by the incorporate the large strain isotropic hardening viscoplastic- optimization formulation and the sensitivity analysis given ity constitutive model, proposed by Simo and Miehe (1992), in Section 6. The paper is closed by presenting several Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2503 numerical examples of optimized composite designs and 3 Constitutive model drawing conclusions. Following Simo and Miehe (1992) and Ivarsson et al. (2018), the material model is based on isotropic hypere- 2 Preliminaries lasticity and isotropic hardening viscoplasticity. For finite strains, the infinitesimal deformation theory’s additive split For nonlinear kinematics, the current, i.e., deformed of the strain tensor is replaced by the finite deformation mul- configuration needs to be distinguished from the tiplicative split F = F eF vp ,whereF e and F vp are the reference, i.e., undeformed configuration o. The location elastic and viscoplastic parts, respectively. of a material particle, which initially at time t = 0 is located The isotropic material response is modeled in terms e e e T at X ∈ o, is described by the smooth motion ϕ such that its of the left Cauchy-Green tensor b = F (F ) , and its e position at time t>0isx = ϕ(X,t). The local deformation isochoric part, b¯ = (J e)−2/3be,whereJ e = det(F e). surrounding X is described by the deformation gradient The hyperelastic response is governed by the neo- F = ∇oϕ = 1 +∇ou,where∇o is the gradient with Hookean strain energy function (see, e.g., Simo and Hughes respect to the material coordinates, X,andu = ϕ −X is the (1998)) displacement field. The volumetric change is described by = 1 1 e the Jacobian J det(F )>0, where the inequality ensures we(be) = κ (J e2 − 1) − 2ln(J e) + μ tr(b¯ ) − 3 , the mapping ϕ is locally unique. The local deformation 4 2 is also described by the right Cauchy-Green deformation (4) tensor C = F T F , and the Green-Lagrange strain tensor where κ and μ are the bulk and shear moduli. Under the E = 1 (C−1). Since the constitutive model is formulated in 2 hyperelastic assumption, the Kirchhoff stress is obtained via an incremental format, we also introduce the spatial velocity e e − τ = 2(∂we/∂b )b . gradient l = FF˙ 1,whereF˙ = dF /dt. The rate-dependent evolution laws are obtained by In absence of body forces and inertia, the momentum maximizing the regularized dissipation balance laws, i.e., equilibrium equation is given by 1 D =˙wvp − Kα˙ − f 2 (5) ∇o · (FS) = 0 in o, (1) 2η ˙ vp = : − 1 L e e−1 where S is the symmetric second Piola-Kirchhoff stress where w τ 2 b b is the viscoplastic power − − tensor, related to the Cauchy stress σ via S = J F 1σF T . per unit volume in the reference configuration, and Lbe = e However, when formulating the constitutive model, we will b˙ − lbe − belT is the Lie derivative of be.In(5), α is utilize the Kirchhoff stress τ that is connected to S and σ the internal variable that defines the isotropic hardening, K through τ = FSFT = J σ. defines the drag stress, η ∈ (0, ∞) is the material viscosity, The boundary conditions associated with (1)are and the Macaulay brackets · are defined such that x = x for x ≥ 0andx = 0forx<0. The von Mises-type ¯ FSNo = t on ∂ot , u = u¯ on ∂ou, (2) dynamic yield function (cf. Ristinmaa and Ottosen 2000)

dev 2 where N o is the outward unit normal vector to the boundary f(τ,K)=||τ || − σy(K) (6) ¯ 3 ∂o of the reference body o and the prescribed traction t and displacement u¯ are defined on the two complimentary separates elastic response, f<0, from viscoplastic dev parts of ∂o, ∂ot and ∂ou, respectively. response, f ≥ 0. In the above (6), τ is the deviatoric part To derive the finite element formulation, we use the of the Kirchhoff stress and principal of virtual work, which can be stated as finding a = + smooth u that satisfies u = u¯ on ∂ou and σy(K) σy0 K (7) is the current yield stress, where σ is the initial yield ¯ y0 R(u,δu) := S : δEdV − t · δudS = 0(3)stress. The drag stress evolves via the nonlinear exponential ∂ o ot hardening model for all smooth admissible virtual displacements δu that K(α) = Hα + Y∞ 1 − exp(−δα) (8) equal zero on ∂ou. The Lagrangian virtual strain δE, = 1 T ∇ + introduced in (3), is defined as δE 2 F oδu where H, Y∞,andδ are the linear hardening modulus, T (∇oδu) F . saturation stress, and saturation exponent, respectively. 2504 N. Ivarsson et al.

Maximizing the regularized dissipation (5) with respect In (12), we introduced the internal force vector to the stresses τ and K provides the evolution equations F int = F int(uˆ , w) and the external load vector F ext ,which are defined by 2 Lbe =− λtr(be)n, (9) = A T F int Be SdV, (13) 3 e o ˙ =− ∂f α λ , (10) = A T ¯ ∂K F ext ξ N e tdS, (14) e ∂ot where n = τ dev/||τ dev|| is the normal to the yield surface, where A is the finite element assembly operator, S = = 1 ˆ ∈[ ] and λ η f is the magnitude of the viscoplastic load S(u, w) and ξ 0, 1 is the load parameter. For simplicity, increment. we drop the superscript (·)(n) and assume that all quantities vp The total amount of absorbed viscoplastic work, W ,is are evaluated at time step tn unless otherwise stated. expressed as 4.2 Integration of constitutive equations T f W vp = w˙ vp dV dt, (11) The residual (12) is evaluated element-wise using Gauss 0 o integration. As such, the constitutive equations need to be integrated in time at each Gauss point of each element in where T f is the load duration. the mesh. To this end, the viscoplastic evolution (9)and(10) are combined with (6)and(7) and discretized in time using the fully implicit Euler scheme. Ultimately we obtain one 4 Numerical solution procedure vector and one scalar local Gauss point-i residual equation. The vector equation is expressed as The boundary value problem defined in (3) is spatially discretized by the finite element method in Section 4.1, 1 e,(n−1) T e 2 (n−1) e Ci = fb f − b − (α − α )tr(b )n = 0, and the constitutive equations are discretized with the 3 backward Euler scheme in Section 4.2. Section 4.3 (15) presents the imposition of the periodic boundary conditions − = −1(n 1) and the nested Newton solution procedure. For further where f FF√ is the relative deformation gradient, =− details of the finite element discretization, see Belytschko ∂f /∂ K 2/3 which follows from (6)and(7), and =−˙ et al. (2000). λ α/(∂f/∂K) which follows from (9). Assuming that viscoplastic response occurs, so f = f ≥ 0 enables ˙ = f ∂f 4.1 Discretization of the equilibrium equations (10) to be expressed as α η ∂K , which gives the scalar discretized residual equation

To obtain the FE-formulation, we approximate the dis- 3 η e 2 = − − (n−1) = placement in each finite element o using the element Ci f(τ,K) α α 0. (16) 2 Δtn shape function matrix, N e, according to u(X,tn) ≈ ˆ (n) ˆ (n) = N e(X)ue ,whereue is the element nodal displacement The residuals (15)and(16) are combined as Ci t 1 T 2 T vector at the discrete time step n. The virtual displace- Ci Ci , forming the Gauss point constitutive resid- ment is likewise interpolated as δu(X) = N e(X)δuˆ e, ual equation i.e., the Galerkin approach is employed. The element ˆ ˆ (n−1) (n−1) = Lagrangian virtual strain is expressed in Voigt notation as Ci(u, u , w, w ) 0. (17) (n) δEe(X,tn) = Be X, uˆ δuˆ e, where the explicit expres- ≥ e Again, assuming f 0, we use (6)and(9) to obtain sion of B is given in Wallin et al. (2018). We also use the e ˙ vp = 2 + 1 T w 3 σy f η f which is combined with (16)to compact internal state variable notation w = beT α , compute the viscoplastic work of (11) where be and all other tensors are expressed in Voigt notation. T f vp = ˙ vp The residual equilibrium (3) is ultimately discretized W w dV dt 0 o as N 3 ≈ σy(α) + ηα˙ αΔt˙ ndV, (18) o 2 t=t R = F int − F ext = 0. (12) n=1 n Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2505

(n−1) where the approximation α˙ = (α − α )/Δtn is where uˇ contains the nodal displacement components of the used. As seen here, the viscoplastic work is a function master and the interior (belonging to o \ ∂o) degrees of of the internal variables from all time steps, i.e., W vp = freedom, and the constant matrix T defines the connection W vp (w, w(n−1),...,w(0)). between the master and slave degrees of freedom. In (20), the vectors I xx and I yx have unit entries respectively for the 4.3 Periodic boundary conditions ux and uy degrees of freedoms corresponding to the nodes x+ on the boundary ∂o and zero entries elsewhere. I xy and We now evaluate the homogenized material properties of the I yy are similarly defined. unit cell microstructure, i.e., composite. As part of this pro- For later purposes, (20) is decomposed into a dependent cess, Bloch-type periodic boundary conditions are applied and independent part. To simplify the derivations, only the to a single unit cell, which implies that the displacement and x uniaxial load case of (19) is considered. To do this, we traction fields on opposite boundaries are periodic, mod- assign the value of u¯xx and decompose (20)as ulo a uniform term and antiperiodic, respectively. Without ∗ uˆ = P uˆ +¯u I , (21) loss of generality, we restrict ourselves to a two-dimensional xx xx (2D) formulation. ∗ T where P = TI I I ,anduˆ = uˇ T u¯ u¯ u¯ The periodicity is based on master-slave elimination xy yx yy xy yx yy x− x+ contains the dependent displacements. To accommodate the wherein the displacement on the boundaries ∂o , ∂o , y− y+ added u¯xy, u¯yx,andu¯yy degrees of freedom, the residuals ∂o ,and∂o (cf. Fig. 1a) are separated into their + − (12)and(17) are supplemented with the null traction master and slave displacements, u and u . Here the first kl kl conditions subscript k represents the displacement component, and the second subscript l represents the boundary, e.g., u+ T = T = T = xy I xyF ext 0, I yxF ext 0, I yyF ext 0, (22) y+ is the ux displacement component over the surface o . y± The differences between master and slave displacement such that the boundaries ∂o are macroscopically traction x± components on opposing faces are, e.g., free and ∂o has zero shear traction, i.e., uniaxial conditions are mimicked. T + − The internal virtual work δWint = F δuˆ together with u¯xx = u − u , (19) int xx xx periodicity of (21) and the null traction of (22) imply that where u¯ is the uniform axial displacement difference = T ˆ ∗ = T T ˇ + ¯ xx δWint F intP δu (T F int) δu Rxyδuxy x+ x− enforced across the boundaries ∂o and ∂o ,cf.Fig.1b. +Ryxδu¯yx + Ryyδu¯yy = 0, (23) The slave degrees of freedom are explicitly eliminated = T from the nonlinear system (12) by expressing the nodal where, e.g., Rxy I xyF int. Equation 23 is used to define displacement vector as the compact residual T ∗ = T T = T uˆ = T uˇ +¯uxxI xx +¯uxyI xy +¯uyxI yx +¯uyyI yy, (20) R (T F int) Rxy Ryx Ryy P F int. (24)

Fig. 1 Illustration of the periodic boundary conditions. a Undeformed unit cell divided into four boundaries and b deformed (dotted lines) unit cell with prescribed and unknown gap displacements, and anti-periodic traction t 2506 N. Ivarsson et al.

∗ The coupled nonlinear system R = 0 and Ci = 5 Regularization and material interpolation 0 of (17)and(24) is solved using two nested Newton loops, wherein linearizations around the current (uˆ , w) In this topology optimization study, we use restriction iterate are performed using a Schur’s complement approach to formulate a well-posed problem. The continuous (Ivarsson et al. 2018;SimoandMiehe1992). At each nondimensional volume fraction field c(X) describes the iteration, an inner Newton loop is first completed to solve amount of material for reference point X, where locations (n) (n−1) (n) (n−1) (n) Ci(uˆ , uˆ , w , w ) = 0 for w for the given filled with material are defined by c(X) = 1andvoid − values of uˆ (n), uˆ (n 1),andw(n−1). The outer loop then locations are identified by c(X) = 0. The continuous commences whence the linearized system volume fraction field c is a function of the design field φ(X) ∈ [0, 1] which is discretized to be piecewise uniform ∗ ˆ ∗ =− ∗ e = ∈ e KT du R (25) over the elements o such that φ(X) φe for X o. In this relationship, c is obtained by smoothing φ so as ˆ ∗ ∗ = T is solved for du ,whereKT P KT P follows from (21) to define a well-posed topology optimization problem. The and (24), in which KT is the consistent tangent stiffness smoothing uses the Helmholtz partial differential equation matrix defined by filtering technique, cf., e.g., Lazarov and Sigmund (2011), wherein −1 ∂F int ∂F int ∂C ∂C 2 K = − , (26) −l Δoc + c = φ, (29) T ∂uˆ ∂w ∂w ∂uˆ o where lo controls the length scale, i.e., smoothing and Δo where C contains the collection of the Ngp Gauss point is the Laplacian with respect to the material coordinates residuals Ci, i = 1, 2,...,Ngp . The use of the consistent X. The filtering technique has previously been used with tangent is crucial for preserving the quadratic rate of homogeneous Neumann boundary conditions, ∇oc·N o = 0 asymptotic convergence of the Newton iteration procedure to conserve volume. In this work, periodicity is enforced on (Simo and Taylor 1985). The successive updates of w(n) and c, which implies that ∇oc · N o is antiperiodic and hence we ∗ uˆ (n) continue until convergence is achieved. The time is also conserve volume. then incremented and the procedure begins anew with the We solve (29) using a standard finite element formula- ˆ ∗(n) tion. The filtered continuous volume fraction field c and initial guess for the displacements u0 . A common choice for the initial displacement guess is the previous converged the weight function δc are interpolated using the element ∗(n) ∗(n−1) c solution, i.e., uˆ = uˆ . Based on our experience, this shape function vector, i.e., c(X) = N e(X)ce within finite 0 e particular choice is not efficient and therefore, we use the element o, and the resulting linear system initial guess Kf c = F f φ (30) (n) is solved for c, i.e., the nodal values of c.In(30), ∗(n) ξ ∗(n−1) uˆ = uˆ , (27) 2 c T c c T c 0 (n−1) Kf = A e (l ∇o N ∇o N e+ N N e)dV is the ξ o o e e = A c T = filter tangent matrix, F f e N e dV ,andφ ∗ o ˆ (n) T i.e., u0 is a scaling of the previous converged solution, [φ1,φ2, ...] is the design variable vector. Since the finite which leads to element discretization is fixed, the matrices Kf and F f are (n) (n) computed once. In order to enforce periodicity, similar to ξ ∗ − ξ − uˆ (n) = P uˆ (n 1) + ξ (n)u¯ I = uˆ (n 1). (28) (20), the constraint 0 (n−1) xx xx (n−1) ξ ξ ∗ c = T f c (31) = ˆ (1) = The first n 1 time step uses the initial guess u0 is included where the connectivity matrix T f connects the (1) ∗ ξ u¯xxI xx. master nodes, stored in c , to the slave nodes. Inserting (31) T into (30), and premultiplying by T f , yields the resulting Remark 1: In the numerical examples, we enforce linear system orthotropic domain symmetry of the unit cell. For the x ∗ ∗ T K c = T F f φ, (32) uniaxial case, this a priori ensures u¯xy =¯uyx = 0and f f R = R = 0. As such, we only enforce the I T F = 0 ∗ = T ∗ xy yx yy ext where Kf T f Kf T f . Upon computing c from (32), the reaction condition to evaluate u¯yy. volume fraction field is obtained from (31) and the finite c element interpolation c(X) = N e(X)ce. Remark 2: For a pure shear load case, we prescribe u¯xy > To approach a distinct solid-void design, we use a ¯ = ¯ ¯ T = 0, uyx 0 and compute uxx and uyy to enforce I xxF ext Heaviside thresholding technique (Guest et al. 2004;Wang T = 0andI yyF ext 0. et al. 2014) and material penalization. First, the continuous Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2507 filtered volume fraction c is thresholded using the smoothed 6.1 Optimization formulation Heaviside step function H¯ , to define the physical volume fraction field The topology optimization problem is to find a design φ by solving tanh(βH ω) + tanh(βH (c − ω)) ⎧ cˆ = H(c)¯ = . (33) ⎪ vp (N) ˆ (0) ˆ ˆ tanh(βH ω) + tanh(βH (1 − ω)) ⎪ max W w (f(φ)),...,w (f(φ)), f(φ) , ⎪ φ ⎧ ⎨⎪ ⎪ N ⎪ 2 β ¯ = O : ⎨ 1 1 (n) ∗ The parameters H and ω are such that limβH →∞ H(c) = ¯ −¯ ≤ ⎪ gu¯yy uyy uyy Δtn 1, − ⎪ s.t. δ ¯ T f us(c ω),whereus is the unit step function. ⎪ ⎪ u = ⎩⎪ ⎩⎪ n 1 Second, the material parameters κ, μ, H, σy0,andY∞ = 1 ˆ ≤ gV¯ ¯ cdV 1, for the constituent material are penalized via the RAMP V o (36) material interpolation scheme (Stolpe and Svanberg 2001). Using the physical volume fraction cˆ of (33), the material where V¯ is the maximum allowable volume in the reference interpolation function for each parameter χ is defined configuration and the mapping as ∗ f(ˆ φ) =ˆc c c(c (φ)) (37) c(ˆ 1 − δ ) χ(cˆ; q,χ ) = δ + 0 χ , (34) 0 0 1 + q(1 −ˆc) 0 which follows from (31)–(33) and the interpolation c = c N ece, within each element. In the above reduced space where the parameter q controls the level of penalization, χ0 optimization problem, we treat the equilibrium (24)and = −4 is the nominal material parameter value, and δ0 10 is local constitutive (17) residuals as implicit constraints, i.e., a small residual value used in this ersatz material model to in (36) we require avoid numerical problems which otherwise arise if, e.g., the material stiffness is zero. ∗ ˆ f R uˆ , w, f(φ) = 0, ∀ tn ∈[0,T ], To summarize, the design variables φe are first filtered ∗ by solving (32)forc , then the thresholded physical (38) periodic volume fraction field cˆ is computed via (31)and (n−1) (n−1) ˆ f C uˆ , w, uˆ , w , f(φ) = 0, ∀ tn ∈[0,T ], (33), and finally the penalized material parameters are (39) evaluated through the application of (34)forχ0 ={κ, μ, H, σy0, Y∞}. ∗ where uˆ = uˆ (uˆ (f(ˆ φ))) and w = w(f(ˆ φ)).

6 Topology optimization Remark 3: The optimization problem (36) allows for maximization of viscoplastic energy absorption subject to a →∞ The objective considered in this topology optimization maximum volume constraint by letting δu¯ . is to design a material microstructure with maximum viscoplastic energy absorption, W vp , cf. (18), subject 6.2 Sensitivity analysis to displacement constraints. One displacement constraint The optimization problem (36) is solved with the MMA enforced during uniaxial loading u¯xx prescribes the scheme, cf. Svanberg (1987), using the standard MMA transverse macroscopic displacement component u¯yy.This Poisson’s ratio-type constraint is imposed by requiring parameters as described in Svanberg (2002). The algorithm uses gradients of the objective and constraint functions to form a convex approximation of the optimization f T 2 1 ∗ problem. We therefore perform a sensitivity analysis in g = u¯ −¯u dt ≤ δ ¯ , (35) u¯yy f yy yy u T 0 each optimization iteration, and as the number of design variables greatly outnumber the constraints, we use the and using the implicit Euler scheme to discretize (35). adjoint method presented in Michaleris et al. (1994). The sensitivity analyses for the objective function W vp and ¯ As seen above, gu¯yy <δu ensures the root-mean-square the constraint gu¯ are performed in a similar manner by deviation (RMSD) of u¯yy from the prescribed target yy ¯∗ · considering the generalized response function trajectory uyy is less than a small value δu¯ taken as 3 −3 10 u¯xx. Similarly, u¯xx can be constrained for a prescribed = uˆ (N),...,uˆ (1), w(N),...,w(0), f(ˆ φ) . (40) uniaxial loading u¯yy. 2508 N. Ivarsson et al.

˜ The sensitivities with respect to the design variable φ are The above is divided into an explicit, DE/Dc,and ˜ ˜ ˜ obtained by multiple applications of the chain rule, i.e., implicit, DI /Dc,partasD/Dc = DE/Dc + ˜ DI /Dc, where the explicit part is ∗ D = D ∂c ∂c ∗ , (41) Dφ Dc ∂c ∂φ N D˜ ∂ T ∂R∗(n) E = − λ∗(N−n+1) Dc ∂c ∂c ∗ n=1 ∂c = ∂c = ∗−1 T cf. (37), where ∂c∗ T f and ∂φ Kf T f Ff N (n) − + T ∂C which follows from (31)and(32), respectively. In (41), − γ (N n 1) (45) · D( ) · = ∂c we introduced D() to denote the total differentiation of ( ) n 1 with respect to (). Differentiation of with respect to the filtered nodal volume fractions c gives and (the transpose of) the implicit part is N ∗(n) D ∂ Duˆ ∂ Dw(n) ∂ = P + + , ˜ T N−1 ˆ ∗(n) T T (n) (n) DI Du T ∂ Dc = ∂uˆ Dc ∂w Dc ∂c =− P − n 1 Dc Dc ˆ (n) n=1 ∂u (42) T ∂F (n) + int ∗(N−n+1) ∗ Pλ where we use the equality ∂uˆ (n)/∂uˆ (n) = P .The ∂uˆ (n) ∗ implicitly defined sensitivities Duˆ (n)/Dc and Dw(n)/Dc T ∂C(n) are annihilated by augmenting a series of adjoint equations + γ (N−n+1) (n) to the functional . To this end, the adjoint vectors λ∗(n) ∂uˆ and γ (n) are used to define the equivalent functional + T N−1 T ∂C(n 1) Dw(n) + γ (N−n) − ˆ (n) Dc ∂u n=1 N N ∗ − + T ∗ − + T T ˜ = − λ (N n 1) R (n) − γ (N n 1) C(n), (43) ∂ T ∂F (n) − + int Pλ∗(N−n+1) n=1 n=1 ∂w(n) ∂w(n) (n) T ∂C − + where we emphasize that the augmented functional ˜ + γ (N n 1) ∂w(n) equals ,since(38)and(39) hold. ⎤ Differentiating the augmented function (43) with respect (n+1) T ∂C (N−n)⎦ to c and applying the chain rule yields + γ ∂w(n) ∗(N) T T ˜ N ∗(n) (n) Duˆ ∂ D ∂ Duˆ ∂ Dw ∂ − P T − = P + + Dc ˆ (N) Dc ∂uˆ (n) Dc ∂w(n) Dc ∂c ∂u n=1 ⎤ (N) T T N (n) ∗ (N) T ∂F ˆ (n) ∂F int ∗(1) ∂C (1)⎦ − ∗(N−n+1) T int Du + Pλ + γ λ P P ˆ (N) ˆ (N) ∂uˆ (n) Dc ∂u ∂u n=1 T ∂F (n) Dw(n) ∂R∗(n) Dw(N) ∂ T +P T int + − − ∂w(n) Dc ∂c Dc ∂w(N) ⎤ N ∗ T T T (n) ˆ (n) (N) (N) (N−n+1) ∂C Du ∂F ∗ ∂C − γ P + int Pλ (1) + γ (1)⎦ . ˆ (n) Dc (N) (N) n=1 ∂u ∂w ∂w ∗ − ∂C(n) Dw(n) ∂C(n) Duˆ (n 1) (46) + + P (n) (n−1) ∂w Dc ∂uˆ Dc (n) (n−1) (n) ∂C Dw ∂C ∗ + + . (44) The implicitly defined derivatives Duˆ (N)/Dc and (n−1) ∂w Dc ∂c Dw(N)/Dc in (46) are annihilated by making judicious Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2509 choices of the adjoint vectors. We first require that λ(1) and Alternatively, substituting (52)into(51), rearranging and γ (1) satisfy using (26), we solve ∗ − + T ∗ ∗ − + K (N n 1) λ (n) =− (N n 1), (53) ⎡ ⎤ T T T (N) (N) T T ⎣ ∂F int ∗(1) ∂C (1)⎦ (N−n+1) T P Pλ + γ ∂C (n) ∂ ˆ (N) ˆ (N) γ = ∂u ∂u ∂w(N−n+1) ∂w(N−n+1) ⎤ T T − + T T ∂ (N−n+2) (N n 1) = P , ∂C − ∂F ∗ (N) (47) − (n 1) − int (n)⎦ ∂uˆ − + γ − + Pλ ∂w(N n 1) ∂w(N n 1) T (N) (N) T T ∂F ∗ ∂C ∂ int Pλ (1) + γ (1) = . (54) ∂w(N) ∂w(N) ∂w(N) (48) where ⎧ ⎨ T −T ∂C(N−n+1) ∂C(N−n+1) ∗(N−n+1) = P T ⎩ ˆ (N−n+1) (N−n+1) Or,byinserting(48)into(47) and some rearrangement, ∂u ∂w ⎡ satisfy T (N−n+2) T × ⎣ ∂ − ∂C ∂w(N−n+1) ∂w(N−n+1) ⎡ T T (N) T ∗ ∗ ∂ ∂C ∂ (K (N))T λ (1) = P T ⎣ − × (n−1) − T (N) (N) γ − + ∂uˆ ∂uˆ ∂uˆ (N n 1) ⎤ ⎫ − T ⎬ (N) T T (N−n+2) ∂C ∂ ⎦ + ∂C (n−1) × , (49) − + γ (55) ∂w(N) ∂w(N) ∂uˆ (N n 1) ⎭ ⎡ ⎤ T T (N) T (N) is the adjoint load vector at step t − + . Equation 53 is ∂C ( ) ∂ ∂F int ∗( ) N n 1 γ 1 =⎣ − Pλ 1 ⎦ , ∗(n) ∂w(N) ∂w(N) ∂w(N) solved for λ and subsequently used in (54) to obtain γ (n). (50) Once the adjoint vectors λ∗(n) and γ (n) are computed, the ˜ sensitivity D/Dc = DE/Dc is obtained from (45), and D/Dφ is computed via (41). ∗(N) (N) = T ¯ where KT P KT P is the tangent operator at the The sensitivity of the constraint guyy is found from the ∗(1) = = terminal time step tN .Onceλ is obtained from (49), the above derivations by equating gu¯yy and thus ∂/∂c adjoint variable γ (1) is obtained from (50). In the remaining 0, ∂/∂w(n) = 0,and ∗ time steps of the adjoint analysis, we determine the λ (n) (N−n+1) ∗(N−n+1) ∂ 1 (n) Dw Duˆ = = −1/2 ¯(n) −¯∗ T and γ to annihilate and for n (δu¯ gu¯yy ) uyy uyy Δtn1yy, (56) Dc Dc ˆ (n) δ ¯ T f 2,...,N, by solving ∂u u ¯(n) = T ˆ (n) where we used uyy 1yyu , where the vector 1yy has a ¯ ⎡ ⎤ single unit entry corresponding to the uyy degree of freedom − + T T ∂F (N n 1) (N−n+1) of the unit cell and is zero elsewhere, cf. the discussion T ⎣ int ∗(n) + ∂C (n)⎦ P − + Pλ − + γ following (21). ∂uˆ (N n 1) ∂uˆ (N n 1) For the objective function, = W vp , the sensitivity is T (N−n+2) T found by using = T ∂ − T ∂C (n−1) P − + P − + γ (51) ∂uˆ (N n 1) ∂uˆ (N n 1) N ∂ = A + − + T T χσ χH α (N n 1) (N−n+1) ∂c e y0 ∂F ∂C = o int Pλ∗(n) + γ (n) n 1 (N−n+1) (N−n+1) ∂w ∂w + [ − − ] | ˙ c T ¯ ⎡ ⎤ χY∞ 1 exp( δα) t=tn αΔtn N e H dV (57) T (N−n+2) T = ⎣ ∂ − ∂C (n−1)⎦ − + − + γ . (52) which follows from (7), (8), (18)and(34), and where e.g., ∂w(N n 1) ∂w(N n 1) = ˆ; ˆ = ¯ χσy0 χ(c q,σy0).In(57), we also used ∂c/∂c H and 2510 N. Ivarsson et al.

= c T e (n) ∂c/∂c N e within the element o.Forthe∂/∂w The primal viscoplasticity problem is solved with an e(n) computation, ∂/∂b¯ , = 0, n = 1,...,N and adaptive time stepping scheme such that N ∂ ∂K 3 η = v = + αΔt˙ V Δtn+1 Δtn(fiter/Nnewt ) , (62) (n) nd ∂α ∂α 2 Δtn = n=1 o t tn N where Nnewt is the number of Newton iterations that is + + 3 ˙ χσy (α) ηα dV, (58) required to attain convergence at the current time step tn, 2 = n=1 o t tn cf., e.g., Borgqvist and Wallin (2013), and the parameters = + − = fiter and v are chosen as fiter = 4andv = 0.4. The where ∂K/∂α χH χY∞ δ exp( δα) and χσy (α) ∗ ∗ − + + [ − − ] convergence criterion is such that |R T Δuˆ | < 10 10 χσy0 χH α χY∞ 1 exp( δα) . Finally, we note that Nmm and diverging time steps are restarted with Δt + ← ∂/∂uˆ (n) = 0. n 1 Δt + /2. The initial time increment is Δt = T f /500 and The last two terms of the sensitivity D˜ /Dc of (45) n 1 1 E the time increments used to perform the sensitivity analysis are computed in the same manner for both the constraint are consistent with those used to solve the primal problem. = g ¯ and for the objective function = W vp . Using uyy Although the analysis is for two dimensions, we use fully (13), (24), ∂c/∂cˆ = H¯ and ∂c/∂c = cN T within e ,the e o integrated 8-node tri-linear brick elements to discretize the first term of (45) is expressed as 1 mm × 1 mm × 0.01 mm unit cell with 80 × 80 × 1 N ∗(n) elements. Plane strain boundary conditions are enforced ∗ − + T ∂R λ (N n 1 on the z−displacement components. Because of this plane ∂c n=1 strain condition, the z-faces of the designs are necessarily N (n) periodic. = A c T (N−n+1)T (n)T ∂S ¯ N e λe Be H dV (59) The Heaviside threshold parameter is ω = 0.5 and a e ∂cˆ = o n 1 continuation strategy is used for βH wherein its initial value −10 e = ∗ βo = 10 is used for the first 80 MMA optimization where λe contains the o element components of λ Pλ , and ∂S(n)/∂cˆ = F −1(∂τ (n)/∂c)ˆ F −T where iterations, whereafter it is increased by increments of 1 nβ β = every design updates until the maximum value of H ˆ = χκ e 2 − + ¯e,dev 10 is reached. In the first two examples nβ = 10, and ∂τ/∂c ((J ) 1)1 χμb (60) 2 for the optimizations including displacement constraints, = which follows from (4), τ = 2(∂we/∂be)be, τ dev = the continuation is slower with nβ 20. The RAMP e,dev penalization parameter is q = 10 for κ, μ, H,andY∞. μb¯ and (34). The last term of the sensitivity (45)is To avoid large viscoplastic strain in low-density elements, computed via the penalization parameter for the initial flow stress σy0 is N (n) chosen such that q = 2

Fig. 3 Macroscopic normalized forces Fxx/(σy0Ao) versus load u¯xx for the one element model

x-face in the reference configuration, versus load u¯xx for the three different macroscopic engineering strain rates ε˙xx = −15 −1 −1 −1 10 s , ε˙xx = 100 s ,andε˙xx = 500 s . The figure clearly shows the influence of the strain rate effects on the microstructural response.

Fig. 2 Illustration of symmetry boundary conditions for domain with 7.2 Test problem: two simple load cases a 4-fold symmetry and b with 8-fold symmetry We start by designing materials subjected to a slow loading The 4-fold symmetry also simplifies the PDE filter rate ε˙ = 10−15 s−1, without any displacement constraints. analysis as periodic boundary conditions are replaced by We do not enforce domain symmetry in this initial example zero flux conditions on all faces. and we consider two load cases: (a) a uniaxial tensile load The initial design must be nonuniform to avoid initially case where u¯xx = 0.01 mm is prescribed and we solve for uniform sensitivities that inhibit the optimization (Alberdi u¯xy, u¯yx, u¯yy,and(b) a pure shear load case where u¯xy = and Khandelwal 2019;Traff¨ et al. 2019). The optimization 0.01 mm and u¯yx = 0 mm are prescribed and we solve for algorithm is therefore initiated such that all design variables u¯xx and u¯yy. Figure 4 shows two optimized microstructural = are set to φe 0.4, followed by a perturbation δ.We designs, both initiated with δ4 = 0.6, i.e., the four consider two different perturbations; (a) a value δ4 is added center elements are assigned the value 1. The yellow and to the four center elements, i.e., φe = 0.4 + δ4, and (b) blue regions represent the base steel material and void, a different random number perturbation is added to all respectively, and we emphasize that the physical volume 1 elements, such that φe = 0.4 + δrnd × (rand(0, 1) − 0.5). fraction field cˆ is plotted. As expected, the two structures dissipating maximum energy consist of bars aligned with 7.1 Constitutive model the maximum principal stress axes. Clearly, these designs are periodic, and although not explicitly enforced, they Prior to the optimization, we demonstrate the rate effects respectively exhibit 4-fold and 8-fold domain symmetries. associated with the constitutive model. The demonstration is performed using a single element model of volume fraction 7.3 Biaxial tensile loads field φe = 1, by simulating the uniaxial response for u¯xx = 0.05 mm with symmetry boundary conditions of In many practical applications, it is of great importance Fig. 2a. Figure 3 shows the macroscopic force component that structures can withstand more complex load cases than T = I xxF int Fxx, normalized by a factor of 1/(σy0Ao), those of uniaxial tension and pure shear. Therefore, we = 2 where Ao 0.01 mm is the area of the loaded unit cell consider a biaxial loading scenario, u¯xx =¯uyy = 0.01 mm again with ε˙ = 10−15 s−1,cf.Fig.5. In this study, we use 1We use Fortrans’ “RANDOM NUMBER,” based on George 4-fold domain symmetry and the corresponding symmetry Marsaglia’s KISS (Keep It Simple, Stupid). boundary conditions of Fig. 2a. 2512 N. Ivarsson et al. a a

b b

c Fig. 4 Undeformed optimized unit cells (left) and array of 3 × 3 unit cells (right)

We present optimized microstructures obtained using three different initial designs. Figure 6 a and b show two designs optimized with δ4 = 0.1 and δ4 =−0.1, respectively. The topology of Design 6a consists of perpendicular bars aligned in the loading directions, whereas Design 6b has a more complex microstructure with both rectangular and elliptical material regions. This difference is reflected in the performance as Design 6a absorbs 1.0 % more viscoplastic Fig. 6 Undeformed optimized unit cells (left) and array of 3 × 3 unit energy than Design 6b, (cf. Fig. 7), i.e., Design 6b is a local cells (right) maxima. We also note that Design 6a absorbs approximately 123 % more specific viscoplastic energy than a fully solid unit cell in which cˆ = 1. Next, the optimization is performed using a random initial 8-fold symmetric perturbation δrnd = 0.1. The optimized design, cf. Fig. 6c, consists of an octagonal frame with connected bars aligned in the loading directions. The energy absorption of this design is also slightly (1.1 %) lower than that of Design 6a. Designs 6a–c are alternate solutions to the optimization problem. And as shown in Fig. 7, the resulting energy absorption capabilities are not equal. In the upper part of this figure, we also see that the convergence towards a distinct solid-void design is slower for Designs 6b–c as compared with Design 6a. It is also seen that only minor design changes are observed when the thresholding parameter βH is increased after optimization iteration 80, but the viscoplastic work increases as the intermediate values of cˆ are thresholded closer towards 0 and 1. We conclude that that the initial design choice is crucial, and Fig. 5 Illustration of the biaxial load case that several local maxima to this optimization problem exist. Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2513

a a

Fig. 7 Topology evolutions of Designs 6a–6c (top) and normalized d vp viscoplastic work W /(σy0Vo) versus optimization iteration, corresponding to the solid, solid-star, and solid-circle lines, respectively (bottom)

7.4 Varying biaxial load paths

Next, we demonstrate the importance of the path-dependent material model for material design under biaxial load cases. We consider the same final load state u¯xx =¯uyy = 0.01 mm e at t = T f as for the previous optimization case, but vary the load path. Specifically, we use the same initial design δ4 = 0.1 and compare the optimized Design 6a, obtained with the f loading conditions u¯xx(t) =¯uyy(t) = t/T · 0.01 mm,to several other loading scenarios with u¯xx(t) =¯uyy(t) for 0

Table 1 a Parameter values for nonproportional biaxial loads gu¯yy is included in the optimizations with the target tra- ∗ f jectory u¯ = 0 ∀ tn ∈[0,T ]. To avoid constraining Design u¯xx u¯yy yy both the transverse and z-component displacements, plane 8b 0 0.020 stress conditions are used, i.e., periodicity is not enforced 8c 0.004 0.020 on the z-faces. An initial design with δrnd = 0.002 is 8d 0.004 0.018 used. For the cases where viscous effects are studied, i.e., − − 8e −0.006 0.014 when ε˙ = 10 15 s 1, the loading rate is ramped up from ε˙ = 0.5 s−1 to the specified final value during the first two optimization iterations. This avoids the possibility that the initial design response is nearly elastic which we found path-dependent model for maximation of viscoplastic otherwise hinders the optimization. energy in microstructural design. 7.5.1 Inﬂuence of strain rate and load magnitude 7.5 Displacement constraints In this example, we design material microstructures that In the following examples, we design microstructures with perform equally under two uniaxial loads u¯xx and u¯yy by nearly zero transverse contraction when subjected to uniax- imposing the 8-fold domain symmetry illustrated in Fig. 2b. ¯ ial tensile loading, e.g., for the uxx loading, the constraint Thus, the computational effort is eased as only the single uniaxial load case u¯xx needs be simulated. The effect of the load magnitude and loading rate on the microstructural a topology is demonstrated by performing five different optimizations (Figs. 9 and 11). In the first three Designs 9a–9c are optimized under the moderate loads u¯xx = 0.01 −15 −1 mm, with slow ε˙xx = 10 s , intermediate ε˙xx = 100 −1 −1 s ,andfastε˙xx = 500 s load rates (cf. Section 7.1 and Fig. 3). We compare the performance of these three designs by exposing them to the same load magnitude u¯xx = 0.01 mm −1 and fast loading rate ε˙xx = 500 s , and measure their corresponding transverse contraction −¯uyy,cf.Fig.10. b

Fig. 10 Contraction −¯uyy versus load u¯xx for Designs 9a–c under −1 the same u¯xx = 0.01 mm load case with ε˙ = 500 s . Top two curves: responses of Designs 9a and 9b, which are not optimized for the fast × ˙ = −1 ≤ Fig. 9 Undeformed optimized designs (left) and array of 3 3 unit ε 500 s load rate. Both Design 9a and Design 9b violate gu¯yy 0. ¯ = ≤ cells (right) under the load uxx 0.01 mm Bottom curve: response of Design 9c which satisfies gu¯yy 0 Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2515

In this figure we see that, although Designs 9a and for a higher loading rate is exposed to higher macroscopic 9c have a similar microstructure, Design 9a violates the forces. contraction constraint (35), cf. the solid-triangle line in Next, we simulate the responses of Designs 11a and 11b Fig. 10. Design 9b, corresponding to the solid-circle line at the load u¯xx = 0.05 mm, but with interchanged loading of Fig. 10, also contracts more than the allowed tolerance rates, i.e., Designs 11a and 11b are subjected to the load rates −3 −5 −1 −15 −1 δu¯ = 3 · 10 u¯xx =3· 10 mm. Figure 10 also shows ε˙ =500 s and ε˙ =10 s , respectively. Results of these that the contraction of all three designs is linearly increasing simulations are displayed with red lines in Fig. 12 aandb. for loads u¯xx less than approximately 0.002 mm but is The bottom two curves of Fig. 12b present the force- nonlinear for higher loads u¯xx > 0.002 mm. Clearly, the displacement responses of Designs 11a and 11b when linear/nonlinear contraction versus load is a consequence of subjected to the slow loading rate. As expected, Design 11b elastic and viscoplastic material response. which is not optimized for this load rate performs worse The last two optimizations are performed for a higher load than Design 11a; it supports less force and violates the −15 −1 u¯xx = 0.05 mm,atslowε˙xx = 10 s and fast ε˙xx = 500 displacement constraint, in fact it has negative transverse s−1 loading rates. The optimized designs are shown in contraction. Surprisingly, Design 11a supports more force Fig. 11. Although no obvious similarities can be seen from the optimized unit cells (cf. the left column of Fig. 11), these two optimizations generate two microstructures with similar a topology as seen in the right column of Fig. 11. Both Designs 11a and 11b satisfy the displacement constraint as shown by the center solid-circle and solid-triangle lines of Fig. 12a. We further analyze the performance of the two Fig. 11 designs by plotting their corresponding macroscopic force- displacement curves, cf. Fig. 12b. The two center solid- circle and solid-triangle lines correspond to the macroscopic force-displacement responses of Designs 11a and 11b, respectively. As expected, Design 11b which is optimized

Fig. 12 Performance of Designs 11a and 11b. a contraction −¯uyy versus load u¯xx and b normalized macroscopic force Fxx/(σy0Ao) Fig. 11 Undeformed optimized designs (left) and array of 3 × 3 unit versus load u¯xx. The red lines with filled markers correspond to cells (right) under the load u¯xx = 0.05 mm responses obtained from simulations with interchanged loading rates 2516 N. Ivarsson et al. and hardens more than Design 11b when it is exposed to However, Design 11a fails to satisfy the displacement a fast loading rate. However, it violates the displacement constraint when subjected to the larger load, cf. Fig. 13b, constraint. These simulations show that more contraction which shows strain-dependent contraction in the u¯xx ∈ leads to higher viscoplastic energy absorption. They also [0.01, 0.05] mm interval, in contrast to Design 11a that demonstrate that it is necessary to include rate effects in the exhibits nearly zero contraction throughout the load optimization of viscoplastic microstructures. interval. Similar to the results presented in Fig. 12,this We also compare the performance of Designs 9a and 11a comparison shows that higher contraction leads to increased by exposing them to the same load u¯xx = 0.05 mm and viscoplastic energy absorption. −15 −1 slow ε˙xx = 10 s load rate. Figure 13 ashowsthat Design 9a which is optimized for the moderate load level 7.5.2 Multiple uniaxial tensile loads: influence of load magnitude u¯xx = 0.01 mm performs slightly better than Design 11a which is optimized for the u¯xx = 0.05 mm load, i.e., in this In this final example, we optimize the unit cell under multiple example, the energy absorption capability is not improved load cases. Specifically, both longitudinal and transversal when optimizing for high load magnitudes. load cases are simulated using 4-fold domain symmetry with symmetry boundary conditions (cf. Fig. 2a). We include

both the gu¯yy and gu¯xx constraints in the optimization. a The objective is to maximize the geometric mean of the vp vp 1/2 vp viscoplastic energy absorption, (Wx · Wy ) ,whereWx vp and Wy are the energies absorbed under the uniaxial loads

b b

Fig. 13 a Normalized viscoplastic energy absorption versus load u¯xx and b −¯uyy versus u¯xx, for Designs 11a and 9a. Design 9a, optimized for u¯xx = 0.01 mm, absorbs 2.7 % more viscoplastic energy but also ¯ ¯∗ = has 6.5 times higher RMSD of uyy from the target uyy 0 during the u¯xx = 0.05 mm load than Design 11a (which is optimized for Fig. 14 Undeformed optimized designs (left) and array of 3 × 3 unit u¯xx = 0.05 mm) cells (right) Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2517

Fig. 17 Normalized macroscopic forces F /(σ A ) and Fig. 15 Normalized macroscopic forces Fxx/(σy0Ao) and xx y0 o F /(σ A ) versus loads u¯ and u¯ for Designs 14a and 14c Fyy/(σy0Ao) versus load u¯xx and u¯yy for Designs 14a–c yy y0 o xx yy

u¯xx and u¯yy, respectively. The optimization is performed 14a supports higher forces and thus can dissipate more for the load magnitudes 0.01 mm,0.05mm, and 0.10 mm viscoplastic energy than Designs 14b and 14c in the load ˙ = −15 −1 with the slow loading rate ε 10 s , all initiated with interval [0, 0.01] mm, for both the u¯xx and u¯yy loads. It the same design. The optimized designs are illustrated in is also seen that Design 14b supports higher forces than Fig. 14. The microstructural Design 14a, optimized for the Design 14c in the load interval [0, 0.05] mm for both loads. moderate load magnitude 0.01 mm, consists of a rectangular Finally, we compare Designs 14a and 14c, optimized for pattern of material with slight offsets to satisfy the near the load magnitudes 0.01 mm and 0.10 mm, respectively, by zero contraction constraints, whereas Designs 14b and 14c, exposing them to the same high load interval [0, 0.10] mm. optimized for the higher load magnitudes 0.05 mm and 0.10 Figure 16 shows pronounced difference in the contraction mm, respectively, consist of more complex topologies. All versus load, e.g., −¯uyy versus u¯xx between Designs 14a three designs satisfy the displacement constraints. and 14c. Indeed, Design 14a has nearly zero contraction in The macroscopic force-displacement responses of the [0, 0.01] mm load interval for which it was designed Designs 14a–14c, plotted in Fig. 15, show that Design but shows a considerable strain-dependent contraction in the (0.01, 0.10] mm load interval. Figure 17 shows the normalized macroscopic force- displacement responses for Designs 14a and 14c. Here it is seen that Design 14c, optimized for the high load magnitude 0.10 mm, softens for loads u¯xx, u¯yy > 0.08 mm, whereas Designs 14a exhibits macroscopic hardening. Similar to the previous studies, this example demonstrates that higher contraction (cf. Fig. 16) leads to more supportive structures (cf. Fig. 17) and thus higher viscoplastic energy absorption.

8 Conclusions

We have established a topology optimization framework for designing periodic microstructural composite materials for maximum viscoplastic energy absorption. Constraints have also been imposed on designs such that, under uniaxial

Fig. 16 Contraction −¯uyy and −¯uxx versus load u¯xx and u¯yy for tensile loads, their macroscopic transverse contraction is Designs 14a and 14c close to zero. The optimization problems are solved using 2518 N. Ivarsson et al. the MMA scheme and sensitivities required to drive the intended use is not permitted by statutory regulation or exceeds algorithm are obtained using the adjoint method. The the permitted use, you will need to obtain permission directly from effects of strain rate, load magnitude, and path-dependency the copyright holder. To view a copy of this licence, visit http:// creativecommonshorg/licenses/by/4.0/. are shown by optimizing designs under several complex macroscopic load cases. It is clear that these effects are of great importance on the optimized microstructural design. Appendix: Sensitivity veriﬁcation The present approach allows each material point in a macroscopic structure to be tailored to the loading rate, load To verify correct implementation of the adjoint sensitiv- path, and load magnitude it experiences. ity analysis presented in Section 6.2, we compare the The unit cell that characterizes the design domain has sensitivities obtained by the adjoint method with numeri- been discretized with 3D finite elements. This formulation cal sensitivities obtained by the central difference method enables a smooth transition to large-scale 3D problems by (CDM). We use 4-fold domain symmetry and the cor- relaxing of the plane strain and plane stress assumptions. responding symmetry boundary conditions of Fig. 2ato In future work, we will apply parallel programming to the discretize a quadrant of the unit cell by 5 × 5 × 1 framework to efficiently solve such large-scale problems. tri-linear brick elements, and subject it to biaxial loading conditions (cf. Fig. 5) with load path illustrated in Acknowledgments The authors would like to thank Prof. Krister Fig. 8e. To ensure large deformations and viscoplastic Svanberg for providing the MMA code. response, the final prescribed displacements u¯xx and u¯yy are increased to 0.05 mm and the load duration is decreased f = Funding information Open access funding provided by Lund Uni- to T 0.01 s; this is done by scaling the Fig. 8 load versity. This work was partially performed under the auspices of the path. Each design variable value is given by the random US Department of Energy by Lawrence Livermore Laboratory under number generator through φe = rand(0, 1), and for the PDE contract DE-AC52-07NA27344, cf. ref number LLNL-CONF-717640. filtering, we set lo = 0.25 mm. The adaptive time-stepping The financial support from the Swedish research council (grant nbr. 2015-05134) and the Swedish Energy Agency (grant nbr. 48344-1) is is omitted for this analysis that instead uses 40 equally gratefully acknowledged. spaced load increments. Figure 18 shows that the sensitivities obtained by the Compliance with ethical standards adjoint (solid, blue curve) and the central difference method (dashed, red curve) are almost equal; the relative error between the two methods is smaller than the perturbation Conﬂict of interest The authors declare that they have no conflict of = −6 interest. size Δh 10 as shown by the dotted line; hence, the derivation and implementation of the adjoint sensitivities are correct. Replication of results The results presented in this article are produced from our Fortran in-house code which uses the Fortran implementation of Prof. Krister Svanbergs MMA optimization solver. We use the Intel Fortran Composer XE 2011 compiler, direct sparse Intel MKL PARDISO solver and utilize multithreading with OpenMP. The geometry input file is generated in Abaqus 6.13. To encourage the replication of the results in this paper, a detailed pseudocode of the optimization framework is given in Algorithms 1 and 2. The data for the initial designs used to generate the results in Section 7.5 and in the Appendix are not included, as they are generated by the pseudorandom number generator and are dependent on the time seeds.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material Fig. 18 Comparison of sensitivities obtained by the adjoint method is not included in the article’s Creative Commons licence and your and the central difference method (CDM) Topology optimization for designing periodic microstructures based on ﬁnite strain viscoplasticity 2519 2520 N. Ivarsson et al.

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