Finite strain topology optimization with nonlinear stability constraints

https://doi.org/10.1016/j.cma.2023.116119Get rights and content

Abstract

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm — the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching, and multiple bifurcations are also demonstrated.

Introduction

Starting from the seminal paper by Bendsœ and Kikuchi [1] introducing homogenization-based topology optimization, the development of computational methods for structural topology optimization has undergone tremendous progress [2]. Besides the linear elastic structures on which the vast majority of the research is still focused [3], the extension to inelasticity [4], [5], finite deformations [6], [7], [8], and multi-physics [9] has also been addressed. To ensure that the optimized structures can operate under the defined loading conditions without failure, the design optimization must, accordingly, incorporate appropriate failure mechanisms. Moreover, the meaning of failure can be quite diverse depending on the context. For example, the failure can be related to the loss of structural stability [10], the emergence of plasticity in the material [11], and the degradation of the material mechanical properties [12], [13], among others. In this study, attention is focused on structures with hyperelastic materials that can sustain large strains within the elastic domain. Accordingly, the failure mechanism is defined as a “buckling-type” instability and for conservative systems, the energy criterion can be used for investigating stability [14], as compared to non-conservative systems which may require a more general Lyapunov criterion [15].

For elastic solids, the most commonly used optimization formulation is minimum compliance subject to material volume constraint also referred to as the stiffness design formulation [2], [6]. For stiffness design, the optimization process may generate slender members, and the stability considerations are then important in producing meaningful practical designs, especially under large deformations [16]. In past, the majority of the topology optimization studies with buckling constraints were devoted to truss structures, mostly limited to linear truss structures, with constraints ranging from simple local buckling constraints using the Euler buckling criterion on each member [17] to the global buckling constraints on the entire structure [18]. The extension to geometrically nonlinear buckling-constrained truss topology optimization was carried out by Li and Khandelwal [19]. In contrast to the truss structures, there are fewer studies on the topology optimization of continuum structures with buckling constraints. For continuum, the first work was carried out by Neves et al. [20], where a linear buckling constraint was incorporated. The linear buckling analysis can be written as a generalized eigenvalue problem K0λKσϕ=0where K0 is the initial stiffness matrix, i.e., at zero displacements, Kσ is the stress stiffness matrix that is evaluated from the displacement field u solved from the linear equations K0u=P where P is the applied force. The smallest eigenvalue is denoted as λ1 represents the buckling load factor that gives the approximate buckling load λ1P, while the corresponding eigenvector ϕ1 represents the buckling mode. It is noted that the linear buckling analysis in Eq. (1) implicitly assumes that the stress stiffness depends linearly on the loads and that the displacements at the critical point are small. Furthermore, for incorporating linear buckling constraints in a density-based topology optimization framework two main challenges have been identified: (a) the presence of spurious buckling modes in the low-density regions [10], [20]; (b) non-differentiability of multiple/repeated eigenvalues λ [21].

In Neves et al. [20], spurious buckling modes were suppressed by ignoring the contributions from the low-density elements to the stress stiffness matrix, and the non-differentiability of multiple eigenvalues is circumvented using the generalized gradient concept. Later, in Bendsœand Sigmund [2], different penalization schemes for the initial stiffness and stress stiffness matrices were proposed to handle the spurious buckling mode issue. Bruyneel [22] demonstrates the necessity of including enough representative buckling modes to handle the switching of buckling modes to avoid poor convergence in linear buckling optimization. The derivatives of repeated eigenvalues have been extensively studied by Seyranian et al. [21] for structural optimization, see also Refs. [23], [24]. The differentiability of the symmetric polynomials of the repeated eigenvalues has been employed in [25] for tackling the non-differentiability related to repeated eigenvalues. Other important contributions to the linear buckling topology optimization can be found in [26], [27], [28]. Recent years have seen an increasing interest in structural buckling topology optimization. For instance, Gao and Ma [29] avoided spurious buckling modes by the eigenvalue shift and mode identification via computing modal strain energy ratio; Dunning et al. [30] used an effective iterative block conjugate gradient method to solve large eigenvalue problems; Thomsen et al. [31] combined linear buckling with Bloch-wave analysis for the design of 2D periodic materials with improved buckling strength; Ferrari et al. [32] studied the use of aggregation functions for approximating the lowest eigenvalue in the buckling constraints, which was adopted by Russ and Waisman [33] for designing elastoplastic structures with damage and linear buckling constraints; Ferrari et al. [34] also developed a multilevel approach for reducing computational cost in large scale optimization problems with linearized buckling constraints; Gao et al. [35] combined linear buckling with stress constraints for the structural stiffness design with improved strength and stability.

Linear buckling analysis is only valid when the deformations are small in the pre-buckling stage and under large deformations, the linear buckling analysis can lead to erroneous results. However, the extension to nonlinear buckling analysis in the topology optimization is not yet fully investigated due to many challenges, e.g., accurate estimation of critical points or other stability indicators, nontrivial sensitivity analysis (even for simple eigenvalues) since the tangent stiffness matrix is a function of the displacement, low-density mesh distortions under large deformations, among others. For instance, Kemmler et al. [36] exploited the so-called extended system of equations to directly calculate the critical point which was constrained from below to minimize compliance. The difficulty is to find the solution to the critical point as the current solution point must be close enough, which is not always guaranteed. As an alternative approach, Lindgaard and Dahl [37] proposed to approximate the critical point at a precritical load step by K0+Kunϕ=λKσnϕwhere Kun and Kσn are the displacement stiffness and stress stiffness at the precritical nth step, respectively. As shown in [19], the accuracy of the approximate critical point depends on the proximity of the precritical step to the critical point. In [36], [37], concerns related to mesh distortions and multiple eigenvalues were not addressed. In a recent study by Dalklint et al. [10], the approach in [37] is extended by adopting a linear energy interpolation scheme [38] to address mesh distortion together with an aggregation function approach for tackling sensitivities of multiple eigenvalues.

On the other hand, as both are concerned with eigenvalue problems, structural eigenfrequency optimization shares common features with structural buckling optimization. It is often formulated as a generalized eigenvalue problem with both mass matrix and stiffness matrices, as compared to the static buckling analysis where only the stiffness matrix is involved. Likewise, most works are confined to linear elasticity. For example, Pedersen [39] maximized the fundamental eigenfrequency in which a new interpolation for stiffness and mass is used in combination with a strategy to neglect nodes surrounded by only low-density elements to avoid spurious vibration modes; Du and Olhoff [40] adopted higher penalization on the mass of low-density elements to suppress spurious eigenmodes, and proposed a two-loop optimization procedure where the inner loop consider directional derivatives to handle the non-differentiability of multiple eigenvalues. This procedure was extended to the geometrically nonlinear case by Dalklint et al. [41]. Besides that, the geometrically nonlinear eigenfrequency optimization was studied by Yoon et al. [42] where the fundamental eigenfrequency of a structure at deformed configuration was maximized in an element-connectivity topological parameterization framework and it was shown that eigenfrequencies of nonlinear systems can be significantly affected by large deformations.

This study focuses on the topology optimization of structures with minimized end compliance while satisfying material volume and nonlinear stability constraints. The main contributions of this work are: (a) A novel strategy for removing spurious buckling modes based on the construction of a pseudo-mass matrix is proposed; (b) A new formulation of nonlinear stability analysis in topology optimization is considered by directly computing the eigenvalues of the tangent stiffness matrix where no other approximations are made; (c) The optimization problem is formulated to incorporate a fixed number of clusters of eigenvalues rather than a fixed number of eigenvalues so that it can handle arbitrary multiplicities of eigenvalues during the optimization process; and (d) Finally, the post-analysis on the B-spline fitted optimized topologies is carried out to evaluate the stability performance of the optimized structures.

The rest of the paper is organized as follows. In Section 2, the density-based framework is briefly reviewed. The nonlinear finite element analysis with a fictitious domain approach used in topology optimization is presented in Section 3. A novel pseudo-mass matrix is developed to handle the spurious buckling modes in the fictitious region in Section 4 together with an illustrative example. Section 5 gives the derivation of the sensitivity analysis for both simple and multiple eigenvalues. In Section 6, a novel optimization formulation that is capable of handling both simple and multiple eigenvalue scenarios is presented. In Section 7, four numerical examples are carried out to demonstrate the effectiveness of the nonlinear buckling constraints in controlling the structural stability under large deformations. Finally, concluding remarks are given in Section 8.

Section snippets

Density-based framework

In the density-based topology optimization [2], with finite element discretization (see Section 3), a design is parameterized by an element-wise constant density field ρX that indicates the presence (ρ=1) or absence (ρ=0) of the material in an element, where XΩ0 denotes an arbitrary material point position in the undeformed reference configuration Ω0. To accommodate gradient-based optimization algorithms, the discrete density variables are relaxed to continuous values, i.e., ρ0,1, where 0<ρ<1

FEA considering finite deformations

Let Ω0R3 be the reference configuration of a deformable continuum body with XΩ0 denoting the position vector of an arbitrary material point in Ω0. It is assumed that a motion that carries the continuum body from its reference configuration to its current configuration ΩtR3 can be described by a smooth one-to-one mapping φ:Xx with uX=φXX, where u represents the displacement field. The associated local deformation gradient is defined by FXφ with JdetF>0, where X denotes the gradient

Nonlinear structural stability analysis with fictitious domain

For a conservative system, an equilibrium state is said to be stable if the potential energy of the system in that state is a proper minimum. After FE discretization with a conforming mesh, the structural stability can be assessed by examining the positive definiteness of the tangent stiffness matrix KT (after the application of boundary conditions). That is, a structure loses stability at a critical point (limit or bifurcation) where the tangent stiffness matrix loses positive definiteness,

Sensitivity analysis of simple/multiple eigenvalues

This section discusses the calculation of the derivatives of eigenvalues w.r.t the design changes as it is an important ingredient in the gradient-based optimization algorithms in topology optimization. When the eigenvalues are simple, the eigenvalues are differentiable, and the calculation details for this case are given in Section 5.1. When the eigenvalues are multiple (repeated), the eigenvalues are not differentiable and only the directional derivatives exist. In this scenario, the

Topology optimization

The goal is to design structures with minimum end compliance for a given amount of material and specified stability objective under the given loads. The stability objective is specified in terms of the constraints on the first (smallest) m different eigenvalues of the tangent stiffness matrix at the final deformation stage. It is noted that m should be large enough to incorporate all the relevant buckling modes for handling mode switching. If the first m eigenvalues are always simple for all

Numerical examples

Before topology optimization, the correctness of the sensitivity analysis is verified. With the derivations given in Section 5, Appendix A Derivatives for the sensitivity analysis of simple/multiple eigenvalues, Appendix B Sensitivity analysis of end compliance and volume fraction, the verification of the sensitivity analysis by the central difference method is given in Appendix C. In the topology optimization, the optimization starts with an initial design consisting of a homogeneous

Conclusions

This study presents a computational framework for geometrically nonlinear topology optimization with nonlinear stability constraints. Unlike the linear buckling constraints that may only provide accurate results when the deformations are small, the nonlinear stability constraints used in the current study provide accurate control over the potential buckling at large deformations. The nonlinear stability analysis at finite deformations is realized with the help of the proposed pseudo-mass matrix

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first author would like to acknowledge the financial support from the National Natural Science Foundation of China (No. 52208467) and the Start-up Research Fund of Southeast University (RF1028623109). The second author would like to acknowledge the support from the US National Science Foundation through grant CMMI-1762277. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

References (65)

  • RussJ.B. et al.

    A novel elastoplastic topology optimization formulation for enhanced failure resistance via local ductile failure constraints and linear buckling analysis

    Comput. Methods Appl. Mech. Engrg.

    (2021)
  • FerrariF. et al.

    Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses

    Comput. Methods Appl. Mech. Engrg.

    (2020)
  • GaoX.

    Improving the overall performance of continuum structures: A topology optimization model considering stiffness, strength and stability

    Comput. Methods Appl. Mech. Engrg.

    (2020)
  • WangF.

    Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems

    Comput. Methods Appl. Mech. Engrg.

    (2014)
  • YoonG.H.

    Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization

    Comput. Struct.

    (2010)
  • ZhouM. et al.

    The COC algorithm, Part II: Topological, geometrical and generalized shape optimization

    Comput. Methods Appl. Mech. Engrg.

    (1991)
  • YoonG.H. et al.

    Element connectivity parameterization for topology optimization of geometrically nonlinear structures

    Int. J. Solids Struct.

    (2005)
  • ZhangG. et al.

    Computational design of finite strain auxetic metamaterials via topology optimization and nonlinear homogenization

    Comput. Methods Appl. Mech. Engrg.

    (2019)
  • ErikssonA.

    Structural instability analyses based on generalised path-following

    Comput. Methods Appl. Mech. Engrg.

    (1998)
  • GrohR.M.J. et al.

    Generalised path-following for well-behaved nonlinear structures

    Comput. Methods Appl. Mech. Engrg.

    (2018)
  • BendsoeM.P. et al.

    Topology Optimization: Theory, Methods, and Applications

    (2003)
  • DeatonJ.D. et al.

    A survey of structural and multidisciplinary continuum topology optimization: post 2000

    Struct. Multidiscip. Optim.

    (2014)
  • KatoJ.

    Analytical sensitivity in topology optimization for elastoplastic composites

    Struct. Multidiscip. Optim.

    (2015)
  • ZhangG. et al.

    Topology optimization of structures with anisotropic plastic materials using enhanced assumed strain elements

    Struct. Multidiscip. Optim.

    (2017)
  • BuhlT. et al.

    Stiffness design of geometrically nonlinear structures using topology optimization

    Struct. Multidiscip. Optim.

    (2000)
  • DalklintA. et al.

    Structural stability and artificial buckling modes in topology optimization

    Struct. Multidiscip. Optim.

    (2021)
  • LiL. et al.

    Failure resistant topology optimization of structures using nonlocal elastoplastic-damage model

    Struct. Multidiscip. Optim.

    (2018)
  • ZhangG. et al.

    Gurson–Tvergaard–Needleman model guided fracture-resistant structural designs under finite deformations

    Internat. J. Numer. Methods Engrg.

    (2022)
  • LiapounoffA.M.

    Probleme General de la Stabilite du Mouvement. (AM-17)

    (2016)
  • ZhangG. et al.

    Topology optimization with incompressible materials under small and finite deformations using mixed u/p elements

    Internat. J. Numer. Methods Engrg.

    (2018)
  • ZhouM.

    Difficulties in truss topology optimization with stress and local buckling constraints

    Struct. Optim.

    (1996)
  • KočvaraM.

    On the modelling and solving of the truss design problem with global stability constraints

    Struct. Multidiscip. Optim.

    (2002)
  • Cited by (6)

    • Topology optimization for maximizing buckling strength using a linear material model

      2023, Computer Methods in Applied Mechanics and Engineering
    • Topology optimization of stability-constrained structures with simple/multiple eigenvalues

      2024, International Journal for Numerical Methods in Engineering
    • Stability constraints for geometrically nonlinear topology optimization

      2023, Structural and Multidisciplinary Optimization
    • Topology Optimization using Nonlinear Finite Element Analysis

      2023, Communications on Applied Nonlinear Analysis
    View full text