The element connectivity parameterization formulation for the topology design optimization of multiphysics systems

In spite of the success of the element‐density‐based topology optimization method in many problems including multiphysics design problems, some numerical difficulties, such as temperature undershooting, still remain. In this work, we develop an element connectivity parameterization (ECP) formulation for the topology optimization of multiphysics problems in order to avoid the numerical difficulties and yield improved results. In the proposed ECP formulation, finite elements discretizing a given design domain are not connected directly, but through sets of one‐dimensional zero‐length links simulating elastic springs, electric or thermal conductors. The discretizing finite elements remain solid during the whole analysis, and the optimal layout is determined by an optimal distribution of the inter‐element connectivity degrees that are controlled by the stiffness values of the links. The detailed procedure for this new formulation for multiphysics problems is presented. Using one‐dimensional heat transfer models, the problem of the element‐density‐based method is explained and the advantage of the ECP method is addressed. Copyright © 2005 John Wiley & Sons, Ltd.     

A new crack tip element for the phantom‐node method with arbitrary cohesive cracks

We have developed a new crack tip element for the phantom‐node method. In this method, a crack tip can be placed inside an element. Therefore, cracks can propagate almost independent of the finite element mesh. We developed two different formulations for the three‐node triangular element and four‐node quadrilateral element, respectively. Although this method is well suited for the one‐point quadrature scheme, it can be used with other general quadrature schemes. We provide some numerical examples for some static and dynamic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Topology optimization of material‐nonlinear continuum structures by the element connectivity parameterization

The application of the element density‐based topology optimization method to nonlinear continuum structures is limited to relatively simple problems such as bilinear elastoplastic material problems. Furthermore, it is very difficult to use analytic sensitivity when a commercial nonlinear finite element code is used. As an alternative to the element density formulation, the element connectivity parameterization (ECP) formulation is developed for the topology optimization of isotropic‐hardening elastoplastic or hyperelastic continua by using commercial software. ECP varies the stiffness of zero‐length linear elastic links that connect design domain‐discretizing finite elements. Unloading was not considered. But the advantages of ECP in material‐nonlinear problems were demonstrated: considerably simple analytic sensitivity calculation using a commercial code and simple link stiffness penalization regardless of nonlinear material behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

P1‐nonconforming quadrilateral finite element for topology optimization

This investigation focuses on an alternative approach to topology optimization problems involving incompressible materials using the P1‐nonconforming finite element. Instead of using the mixed displacement‐pressure formulation, a pure displacement‐based approach can be employed for finite element formulation owing to the Poisson locking‐free property of the P1‐nonconforming element. Moreover, because the P1‐nonconforming element has linear shape functions that are defined at element vertices, it has considerably fewer degrees of freedom than other quadrilateral nonconforming elements and its implementation is as simple as that of the conforming bilinear element. Various problems dealing with incompressible materials and pressure‐loaded structures found in published works are solved to verify the applicability of the proposed method. The application of the method is extended to the optimal design of fluid channels in the Stokes flow. This is done by expressing pressure in terms of volumetric strain rates and developing a velocity‐field‐only finite element formulation. The optimization results obtained from all the problems considered in this study are in close agreement with those found in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A boundary‐perturbation finite element approach for shape optimization

A numerical method is proposed for the efficient solution of shape optimization problems, which combines the boundary perturbation technique and finite element analysis. The method is computationally efficient in that it requires a number of finite element analyses with a fixed geometry, as opposed to standard shape optimization which requires re‐analysis with varying geometry. The application of the method to general shape optimization is considered. In addition, a special optimization scheme is devised for a class of problems governed by linear partial differential equations. The performance of the method is illustrated via an example which involves acoustic wave scattering from an obstacle. Copyright © 2000 John Wiley & Sons, Ltd.     

Sensitivity analysis and design optimization of three‐dimensional non‐linear aeroelastic systems by the adjoint method

We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple approach towards fully stressed design in hyperelasticity exploiting the isoparametric finite element concept

A novel approach towards fully stressed designs in hyperelasticity is discussed leading to closed‐form expressions for the sensitivities of the objective and displacements with respect to design variations. The key idea is the modification of the classical approach coupled with a so‐called design element method offering a lot of parallelism to standard finite element methods. We bypass implicit constraints on dependent quantities and derive an explicit linearly constrained optimization problem solved by means of first‐order procedures. The results obtained with the proposed method are adequate from an engineering point of view though being computed with a simple method. Copyright © 2000 John Wiley & Sons, Ltd.     

Multiphase topology design with optimal material selection using an inverse p‐norm function

We present an original mathematical formulation for optimizing structural topology while simultaneously identifying an optimal set of design materials that are selected from a larger set of candidate materials. This design task is analogous to that, which is commonly encountered in additive manufacturing applications in which the 3D printer can print parts containing up to 3 distinct materials that can be selected from a larger suite of usable materials. The material distribution is parameterized via the shape functions with penalization formulation in which a set of activation functions, which are derived from a partition of the unit hypercube, is used to determine the effective local elasticity modulus within a single finite element. Additionally, we introduce an inverse p‐norm function, which is used to ensure that the optimized material properties converge to a set of discrete values corresponding to the available candidate materials. The algorithm has been implemented on a set of 2D benchmark problems. Numerical results show that the formulation combining the inverse p‐norm function with the activation functions successfully produces optimized multimaterial solutions containing no more than the prescribed number of distinct materials.     

An efficient hybrid algorithm for the optimization of problems with several local minima

The problem of design optimization is of high industrial interest, and has been extensively studied for years, with excellent results. However, there is the well‐known issue of a reasonable balance between the computational effort usually required by stochastic methods, and the fact that deterministic optimizers, even though much more efficient, are not guaranteed to localize a good minimum, as they can remain trapped in the first found local one. To overcome these problems, the authors developed a hybrid strategy, which gave good results in terms of speed and reliability of the obtained optima, especially when the objective function is obtained through a finite element analysis, due, for example, to the absence of an analytical solution of the problem, and the direct use of a stochastic method would be unfeasible for practical purposes, because of the intolerable processing time required. Copyright © 2001 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.     

Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved weighting method with multibounds formulation and convex programming for multicriteria structural optimization

This paper presents an improved weighting method for multicriteria structural optimization. By introducing artificial design variables, here called as multibounds formulation (MBF), we demonstrate mathematically that the weighting combination of criteria can be transformed into a simplified problem with a linear objective function. This is a unified formulation for one criterion and multicriteria problems. Due to the uncoupling of involved criteria after the transformation, the extension and the adaptation of monotonic approximation‐based convex programming methods such as the convex linearization (CONLIN) or the method of moving asymptotes (MMA) are made possible to solve multicriteria problems as efficiently as for one criterion problems. In this work, a multicriteria optimization tool is developed by integrating the multibounds formulation with the CONLIN optimizer and the ABAQUS finite element analysis system. Some numerical examples are taken into account to show the efficiency of this approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Explicit level‐set‐based topology optimization using an exact Heaviside function and consistent sensitivity analysis

This paper presents a level‐set‐based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest‐descent update of the design variables in a level‐set method; the level‐set nodal values. An exact Heaviside formulation is used to relate the level‐set function to element densities. The level‐set function is not required to be a signed‐distance function, and reinitialization is not necessary. Using this approach, level‐set‐based topology optimization problems can be solved consistently and multiple constraints treated simultaneously. The proposed method leads to more insight in the nature of level‐set‐based topology optimization problems. The level‐set‐based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level‐set‐based or density‐based regularization can be used to avoid artifacts in the final results. The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level‐set‐based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Copyright © 2012 John Wiley & Sons, Ltd.     

Control of analyses with isoparametric elements in both 2‐D and 3‐D

This paper presents an extension of the computation method for the error in the constitutive relation to include isoparametric elements. Developed over the past few years at ENS Cachan's laboratory, this technique is based on the strict building of admissible displacement and stress fields. This building process, validated for several types of 2‐D and 3‐D straight finite elements (triangles and quadrilaterals, tetrahedra and hexahedra), cannot be extended to isoparametric elements. For such elements, the method consists of seeking an approximation of the statically admissible stress field by solving a high‐degree finite element problem on each element. This technique, as implemented in our error computation code, which is associated both with a method of computing optimal sizes and with meshers able to respect a size map, allows us to optimize 2‐D and 3‐D meshes. Examples demonstrate the capabilities of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Nitsche's method for Helmholtz problems with embedded interfaces

In this work, we use Nitsche's formulation to weakly enforce kinematic constraints at an embedded interface in Helmholtz problems. Allowing embedded interfaces in a mesh provides significant ease for discretization, especially when material interfaces have complex geometries. We provide analytical results that establish the well‐posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche's method is used to enforce kinematic constraints. As in the analysis of conventional Helmholtz problems, we show that the inf‐sup constant remains positive provided that the Nitsche's stabilization parameter is judiciously chosen. We then apply our formulation to several 2D plane‐wave examples that confirm our analytical findings. Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.