Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Three‐level BDDC in two dimensions

Balancing Domain Decomposition by Constraints (BDDC) methods are non‐overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems. They are similar to the balancing Neumann–Neumann algorithm. However, in BDDC methods, a small number of continuity constraints are enforced across the interface, and these constraints form a new coarse, global component. An important advantage of using such constraints is that the Schur complements that arise in the computation will all be strictly positive definite. The matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation. However, this problem can ultimately become a bottleneck, if the number of subdomains is very large. In this paper, two three‐level BDDC methods are introduced for solving the coarse problem approximately in two‐dimensional space, while still maintaining a good convergence rate. Estimates of the condition numbers are provided for the two three‐level BDDC methods and numerical experiments are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

An evaluation of limited‐memory sparse linear solvers for thermo‐mechanical applications

We consider the performance of sparse linear solvers for problems that arise from thermo‐mechanical applications. Such problems have been solved using sparse direct schemes that enable robust solution at the expense of memory requirements that grow non‐linearly with the dimension of the coefficient matrix. In this paper, we consider a class of preconditioned iterative solvers as a limited‐memory alternative to direct solution schemes. However, such preconditioned iterative solvers typically exhibit complex trade‐offs between reliability and performance. We therefore characterize such trade‐offs for systems from thermo‐mechanical problems by considering several preconditioning schemes including multilevel methods and those based on sparse approximate inversion and incomplete matrix factorization. We provide an analysis of computational costs and memory requirements for model thermo‐mechanical problems, indicating that certain incomplete factorization schemes can achieve good performance. We also provide empirical evaluations that corroborate our analysis and indicate the relative effectiveness of different solution schemes. Our results indicate that our drop‐threshold incomplete Cholesky preconditioning is more robust, efficient and flexible than other popular preconditioning schemes. In addition, we propose preconditioner reuse to amortize preconditioner construction cost over a sequence of linear systems that arise from non‐linear solutions in a plastic regime. Copyright © 2007 John Wiley & Sons, Ltd.     

Subspace recycling accelerates the parametric macro‐modeling of MEMS

A fast computational technique that speeds up the process of parametric macro‐model extraction is proposed. An efficient starting point is the technique of parametric model order reduction (PMOR). The key step in PMOR is the computation of a projection matrix V, which requires the computation of multiple moment matrices of the underlying system. In turn, for each moment matrix, a linear system with multiple right‐hand sides has to be solved. Usually, a considerable number of linear systems must be solved when the system includes more than two free parameters. If the original system is of very large size, the linear solution step is computationally expensive. In this paper, the subspace recycling algorithm outer generalized conjugate residual method combined with generalized minimal residual method with deflated restarting (GCRO‐DR), is considered as a basis to solve the sequence of linear systems. In particular, two more efficient recycling algorithms, G‐DRvar1 and G‐DRvar2, are proposed. Theoretical analysis and simulation results show that both the GCRO‐DR method and its variants G‐DRvar1 and G‐DRvar2 are very efficient when compared with the standard solvers. Furthermore, the presented algorithms overcome the bottleneck of a recently proposed subspace recycling method the modified Krylov recycling generalized minimal residual method. From these subspace recycling algorithms, a PMOR process for macro‐model extraction can be significantly accelerated. Copyright © 2013 John Wiley & Sons, Ltd.     

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Linear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient‐based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level‐set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub‐problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.     

On the convergence of stationary sequences in topology optimization

We consider structural topology optimization problems including unilateral constraints arising from non‐penetration conditions in contact mechanics. The resulting non‐convex non‐smooth problems are instances of mathematical programs with equilibrium constraints (MPEC), or bi‐level programs. Applying nested (implicit programming) algorithms to this class of problems is problematic owing to the singularity of the feasible set. We propose a perturbation strategy combining the relaxation of the equilibrium constraint with the restriction of the design domain to its regular part only. This strategy allows us to attack the problem numerically using standard non‐linear programming algorithms. We rigorously study the optimality conditions for the original singular problem as well as the convergence of stationary points and globally optimal solutions to approximating problems towards respective stationary points and globally optimal solutions to the original problem. A limited numerical benchmarking of the algorithm is performed. Copyright © 2005 John Wiley & Sons, Ltd.     

A nonlinear programming technique to compute a tight lower bound for the real structured singular value

The real structured singular value (RSSV, or real μ) is a useful measure to analyze the robustness of linear systems subject to structured real parametric uncertainty, and surely a valuable design tool for the control systems engineers. We formulate the RSSV problem as a nonlinear programming problem and use a new computation technique, F-modified subgradient (F-MSG) algorithm, for its lower bound computation. The F-MSG algorithm can handle a large class of nonconvex optimization problems and requires no differentiability. The RSSV computation is a well known NP hard problem. There are several approaches that propose lower and upper bounds for the RSSV. However, with the existing approaches, the gap between the lower and upper bounds is large for many problems so that the benefit arising from usage of RSSV is reduced significantly. Although the F-MSG algorithm aims to solve the nonconvex programming problems exactly, its performance depends on the quality of the standard solvers used for solving subproblems arising at each iteration of the algorithm. In the case it does not find the optimal solution of the problem, due to its high performance, it practically produces a very tight lower bound. Considering that the RSSV problem can be discontinuous, it is found to provide a good fit to the problem. We also provide examples for demonstrating the validity of our approach.     

Numerical methods for interactive multiple‐class image segmentation problems

In this article, we consider a bilaterally constrained optimization model arising from the semisupervised multiple‐class image segmentation problem. We prove that the solution of the corresponding unconstrained problem satisfies a discrete maximum principle. This implies that the bilateral constraints are satisfied automatically and that the solution is unique. Although the structure of the coefficient matrices arising from the optimality conditions of the segmentation problem is different for different input images, we show that they are M‐matrices in general. Therefore, we study several numerical methods for solving such linear systems and demonstrate that domain decomposition with block relaxation methods are quite effective and outperform other tested methods. We also carry out a numerical study of condition numbers on the effect of boundary conditions on the optimization problems, which provides some insights into the specification of boundary conditions as an input knowledge in the learning context. © 2010 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 20, 191–201, 2010     

Optimization of large‐scale truss structures using sparse SAND formulations

Sparsity features of simultaneous analysis and design (SAND) formulations are studied and exploited for optimization of large‐scale truss structures. Three formulations are described and implemented with an existing analysis code. SAND formulations have large number of variables; however, gradients of the functions and Hessian of the Lagrangian are quite sparsely populated. Therefore, this structure of the problem is exploited and an optimization algorithm with sparse matrix capability is used to solve large‐scale problems. An existing analysis software is integrated with an optimizer based on a sparse sequential quadratic programming (SQP) algorithm to solve sample problems. The formulations and algorithms work quite well for the sample problems, and their performances are compared and discussed. For all the cases considered, the SAND formulations out perform the conventional formulation except one case. Further research is suggested to fully study and utilize sparse features of the alternative SAND formulations and to develop more efficient sparse solvers for large‐scale and more complex applications. Copyright © 2006 John Wiley & Sons, Ltd.     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

Density gradient-based adaptive refinement of analysis mesh for efficient multiresolution topology optimization

In topology optimization, the finite-element analysis of the problem is generally the most computationally demanding task of the solution process. In order to improve the efficiency of this phase, in this article we propose to represent regions with zero density gradient by a coarser analysis mesh. The design is instead represented in a uniform mesh. We motivate the density gradient-based adaptive refinement by discussing the topological meaning of the density gradient and how it can help avoid loss of information during projections or interpolations between design and analysis meshes. We also study the adaptiveness of the mesh and its ability to detect the topology change of the design. An a posteriori error analysis is performed as well. Furthermore, we provide theoretical and numerical considerations on the reduction of the number of degrees of freedoms of the adaptive analysis mesh with respect to the uniform case. This translates into a faster solution of the analysis, as we show numerically. Finally, we solve several test problems, including large 3D problems that we solve in parallel on computer cluster, demonstrating the applicability of our procedure in large scale computing and with iterative solvers.     

A probabilistic framework for multidisciplinary design: Application to the hydrostructural optimization of supercavitating hydrofoils

The analysis and optimization of complex multiphysics systems presents a series of challenges that limit the practical use of computational tools. Specifically, the optimization of such systems involves multiple interconnected components with competing quantities of interest and high‐dimensional spaces and necessitates the use of costly high‐fidelity solvers to accurately simulate the coupled multiphysics. In this paper, we put forth a data‐driven framework to address these challenges leveraging recent advances in machine learning. We combine multifidelity Gaussian process regression and Bayesian optimization to construct probabilistic surrogate models for given quantities of interest and explore high‐dimensional design spaces in a cost‐effective manner. The synergistic use of these computational tools gives rise to a tractable and general framework for tackling realistic multidisciplinary optimization problems. To demonstrate the specific merits of our approach, we have chosen a challenging large‐scale application involving the hydrostructural optimization of three‐dimensional supercavitating hydrofoils. To this end, we have developed an automated workflow for performing multiresolution simulations of turbulent multiphase flows and multifidelity structural mechanics (combining three‐dimensional and one‐dimensional finite element results), the results of which drive our machine learning analysis in pursuit of the optimal hydrofoil shape.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Enhanced balancing Neumann–Neumann preconditioning in computational fluid and solid mechanics

In this work, we propose an enhanced implementation of balancing Neumann–Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo‐inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo‐inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large‐scale distributed‐memory platforms. The enhanced BNN method is a competitive preconditioner for three‐dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical methods for the topology optimization of structures that exhibit snap‐through

The inclusion of non‐linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response, e.g. the geometric non‐linear response of compliant mechanisms. In previous work, the non‐linear response is computed by standard non‐linear elastic finite element analysis. Here, we incorporate a load–displacement constraint method to traverse non‐linear equilibrium paths with limit points to design structures that exhibit snap‐through behaviour. To accomplish this, we modify the basic arc length algorithm and embed this analysis into the topology optimization problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Topology optimization with multiple materials via moving morphable component (MMC) method

With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.     

A fast method for binary programming using first‐order derivatives,with application to topology optimization with buckling constraints

We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.     

Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficients

Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse‐grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd.