Multimaterial multijoint topology optimization

In this paper, a methodology that solves multimaterial topology optimization problems while also optimizing the quantity and type of joints between dissimilar materials is proposed. Multimaterial topology optimization has become a popular design optimization technique since the enhanced design freedom typically leads to superior solutions; however, the conventional assumption that all elements are perfectly fused together as a single piece limits the usefulness of the approach since the mutual dependency between optimal multimaterial geometry and optimal joint design is not properly accounted for. The proposed methodology uses an effective decomposition approach to both determine the optimal topology of a structure using multiple materials and the optimal joint design using multiple joint types. By decomposing the problem into two smaller subproblems, gradient‐based optimization techniques can be used and large models that cannot be solved with nongradient approaches can be solved. Moreover, since the joining interfaces are interpreted directly from multimaterial topology optimization results, the shape of the joining interfaces and the quantity of joints connecting dissimilar materials do not need to be defined a priori. Three numerical examples, which demonstrate how the methodology optimizes the geometry of a multimaterial structure for both compliance and cost of joining, are presented.     

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.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

Optimization of truss topology using tabu search

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross-sections are assumed to be continuous, the topological optimization problem has mixed discrete-continuous variables. This non-linear programming problem is solved using a memory-based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.     

Voigt–Reuss topology optimization for structures with nonlinear material behaviors

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance. © 1997 John Wiley & Sons, Ltd.     

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.     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the finite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems.Each of them is solved in the dual space with a subgradient‐based algorithm (or with a variant of it) specially developed to maximize the not everywhere differentiable dual function. To show that the application field is large, the presented applications are issued from different domains of structural design, such as sizing of thin‐walled structures, geometrical configuration of trusses, topology optimization of membrane or 3‐D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non‐convexity of the primal discrete subproblems. Fortunately, this gap can be quantified: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very efficient for 2‐D and 3‐D topology optimization, where applications involving thousands of binary design variables are solved in a very short time. Copyright © 2000 John Wiley & Sons, Ltd.     

Robust topology optimization under multiple independent uncertainties of loading positions

The topology optimization problem of a continuum structure is further investigated under the independent position uncertainties of multiple external loads, which are now described with an interval vector of uncertain-but-bounded variables. In this study, the structural compliance is formulated with the quadratic Taylor series expansion of multiple loading positions. As a result, the objective gradient information to the topological variables can be evaluated efficiently upon an explicit quadratic expression as the loads deviate from their ideal application points. Based on the minimum (largest absolute) value of design sensitivities, which corresponds to the most sensitive compliance to the load position variations, a two-level optimization algorithm within the non-probabilistic approach is developed upon a gradient-based optimization method. The proposed framework is then performed to achieve the robust optimal configurations of four benchmark examples, and the final designs are compared comprehensively with the traditional topology optimizations under the loading point fixation. It will be observed that the present methodology can provide a remarkably different structural layout with the auxiliary components in the design domain to counteract the load position uncertainties. The numerical results also show that the present robust topology optimization can effectively prevent the structural performance from a noticeable deterioration than the deterministic optimization in the presence of load position disturbances.     

A scaled boundary finite element based explicit topology optimization approach for three-dimensional structures

This article proposes an efficient approach for solving three-dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)-based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.     

Radial basis functions and level set method for structural topology optimization

Level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel algorithm using an orthotropic material model for topology optimization

This article presents a novel algorithm for topology optimization using an orthotropic material model. Based on the virtual work principle, mathematical formulations for effective orthotropic material properties of an element containing two materials are derived. An algorithm is developed for structural topology optimization using four orthotropic material properties, instead of one density or area ratio, in each element as design variables. As an illustrative example, minimum compliance problems for linear and nonlinear structures are solved using the present algorithm in conjunction with the moving iso-surface threshold method. The present numerical results reveal that: (1) chequerboards and single-node connections are not present even without filtering; (2) final topologies do not contain large grey areas even using a unity penalty factor; and (3) the well-known numerical issues caused by low-density material when considering geometric nonlinearity are resolved by eliminating low-density elements in finite element analyses.     

Multi-material topology optimization of laminated composite beam cross sections

This paper presents a novel framework for simultaneous optimization of topology and laminate properties in structural design of laminated composite beam cross sections. The structural response of the beam is evaluated using a beam finite element model comprising a cross section analysis tool which is suitable for the analysis of anisotropic and inhomogeneous sections of arbitrary geometry. The optimization framework is based on a multi-material topology optimization model in which the design variables represent the amount of the given materials in the cross section. Existing material interpolation, penalization, and filtering schemes have been extended to accommodate any number of anisotropic materials. The methodology is applied to the optimal design of several laminated composite beams with different cross sections. Solutions are presented for a minimum compliance (maximum stiffness) problem with constraints on the weight, and the shear and mass center positions. The practical applicability of the method is illustrated by performing optimal design of an idealized wind turbine blade subjected to static loading of aerodynamic nature. The numerical results suggest that the proposed framework is suitable for simultaneous optimization of cross section topology and identification of optimal laminate properties in structural design of laminated composite beams.     

Multi-objective ceramic matrix composite material design using the variable fidelity model management optimization framework

The increasing computational requirements of advanced numerical tools for simulating material behaviour can prohibit direct integration of these tools in a design optimization procedure where multiple iterations are required. Therefore, a design approach is needed that can incorporate multiple simulations (multi-physics with different input variables) of varying fidelity in an iterative model management framework that can significantly reduce design cycle times. In this research, a material design tool based on a variable fidelity model management framework is applied to obtain the optimal size of a second phase, consisting of silicon carbide (SiC) fibres, in a silicon-nitride (Si₃N₄) matrix to obtain continuous fibre SiC-Si₃N₄ ceramic composites (CFCCs) with maximum high temperature strength and high temperature creep resistance. This investigation shows how models with different dimensions and input design variables can be handled and integrated efficiently by the trust region model management framework, while significantly reducing design cycle times in application to the design of multiphase composite materials.     

A velocity field level set method for shape and topology optimization

In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.     

Structural optimization of uncertain dynamical systems considering mixed-design variables

In this paper attention is directed to the reliability-based optimization of uncertain structural systems under stochastic excitation involving discrete-continuous sizing type of design variables. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as a measure of system reliability. The problem is solved by a sequential approximate optimization strategy cast into the framework of conservative convex and separable approximations. To this end, the objective function and the reliability constraints are approximated by using a hybrid form of linear, reciprocal and quadratic approximations. The approximations are combined with an effective sensitivity analysis of the reliability constraints in order to generate explicit expressions of the constraints in terms of the design variables. The explicit approximate sub-optimization problems are solved by an appropriate discrete optimization technique. The optimization scheme exhibits monotonic convergence properties. Two numerical examples showing the effectiveness of the approach reported herein are presented.     

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.