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.     

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 new formulation and 𝒞0‐implementation of dynamically consistent gradient elasticity

In his article a special form of gradient elasticity is presented that can be used to describe wave dispersion. This new format of gradient elasticity is an appropriate dynamic extension of the earlier static counterpart of the gradient elasticity theory advocated in the early 1990s by Aifantis and co‐workers. In order to capture dispersion of propagating waves, both higher‐order inertia and higher‐order stiffness contributions are included, a fact which implies (and is denoted as) dynamic consistency. The two higher‐order terms are accompanied by two associated length scales. To facilitate finite element implementations, the model is rewritten such that ??⁰‐continuity of the interpolation is sufficient. An auxiliary displacement field is introduced which allows the original fourth‐order equations to be split into two coupled sets of second‐order equations. Positive‐definiteness of the kinetic energy requires that the inertia length scale is larger than the stiffness length scale. The governing equations, boundary conditions and the discretized system of equations are presented. Finally, dispersive wave propagation in a one‐dimensional bar is considered in a numerical example. Copyright © 2007 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.     

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.     

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.     

Optimal layout design of three‐dimensional geometrically non‐linear structures using the element connectivity parameterization method

The topology design optimization of ‘three‐dimensional geometrically‐non‐linear’ continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization. To overcome these difficulties, the element connectivity parameterization (ECP) method with two implementation formulations is proposed. In ECP, structural layouts are represented by inter‐element connectivity, which is controlled by the stiffness of element‐connecting zero‐length links. Depending on the link location, ECP may be classified as an external ECP (E‐ECP) or an internal ECP (I‐ECP). In this paper, I‐ECP is newly developed to substantially enhance computational efficiency. The main idea in I‐ECP is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level. As for ECP implementation with commercial software, E‐ECP, developed earlier for two‐dimensional problems, is easier to use even for three‐dimensional problems because it requires only numerical analysis results for design sensitivity calculation. The characteristics of the I‐ECP and E‐ECP methods are compared, and these methods are validated with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Geometry and topology optimization of sheet metal profiles by using a branch‐and‐bound framework

In this paper well established procedures from partial differential equation (PDE)‐constrained and discrete optimization are combined in a new way to find an optimal design of a multi‐chambered profile. Given a starting profile design, a load case and corresponding design constraints (e.g. sheet thickness, chamber sizes), the aim is to find an optimal subdivision into a predefined number of chambers with optimal shape subject to structural stiffness. In the presented optimization scheme a branch‐and‐bound tree is generated with one additional chamber in each level. Before adding the next chamber, the geometry of the profile is optimized. Then a relaxation of a topology optimization problem is solved. Based on this relaxation, a best fitting feasible topology subject to manufacturability conditions is determined using a new mixed integer method employing shortest paths. To improve the running time, the finite element simulations for the geometry optimization and topology relaxation are performed with different levels of accuracy. Finally, numerical experiments are presented including different starting geometries, load scenarios and mesh sizes.     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient integration technique for the voxel‐based finite cell method

The finite cell method is a fictitious domain approach based on hierarchical Ansatz spaces of higher order. The method avoids time‐consuming and often error‐prone mesh‐generation and favorably exploits Cartesian grids to embed structures of complex geometry in a simple‐shaped computational domain thus shifting parts of the computational effort from mesh generation to the computation within the embedding finite cells of regular shape. This paper presents an effective integration approach for voxel‐based models of linear elasticity that drastically reduces the computational effort on cell level. The applied strategy allows the pre‐computation of an essential part of the cell matrices and vectors of higher order, representing stiffness and load, respectively. Several benchmark problems show the potential of the proposed method in particular for heterogeneous material properties as common in biomedical applications based on computer tomography scans. The applied strategy ensures a fast computation for time‐critical simulations and even allows user‐interactive simulations for models of moderate size at a high level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Large strain phase‐field‐based multi‐material topology optimization

A multi‐material topology optimization scheme is presented. The formulation includes an arbitrary number of phases with different mechanical properties. To ensure that the sum of the volume fractions is unity and in order to avoid negative phase fractions, an obstacle potential function, which introduces infinity penalty for negative densities, is utilized. The problem is formulated for nonlinear deformations, and the objective of the optimization is the end displacement. The boundary value problems associated with the optimization problem and the equilibrium equation are solved using the finite element method. To illustrate the possibilities of the method, it is applied to a simple boundary value problem where optimal designs using multiple phases are considered. Copyright © 2015 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

Integrated layout design of multi‐component system

A new integrated layout optimization method is proposed here for the design of multi‐component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non‐overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal. Finally, to validate the proposed design procedure, a variety of multi‐component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Reliability‐based topology optimization of structures under stress constraints

In this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.