On compliance minimization by biomimetic regularization method

Topology optimization has been successfully used in many industries, especially those engaged in the design and manufacturing of mechanical devices, but numerical problems are often encountered, such as grayscale representations of obtained composites. A type of structural optimization method using the level set theory for boundary expressions has been proposed, in which the outlines of target structures are implicitly represented using the level set function, and optimal configurations are obtained by updating this function based on the shape sensitivities. Level set-based methods typically have a drawback, however, in that topological changes that increase the number of holes in the material domain are not allowed. To overcome the above numerical and topological problems, this paper proposes a new topology optimization method incorporating level set boundary expressions based on the concept of the phase field method, which we apply to a minimum mean compliance problem. First, a structural optimization problem is formulated based on a boundary expression, using the level set function. Next, a time evolutionary equation for updating the level set function is formulated based on the concept of the phase field method, and the minimum mean compliance problem is formulated using a level set boundary expression. An optimization algorithm for the topology optimization incorporating the level set boundary expression based on the concept of the phase field method is then derived. Several examples are provided to confirm the usefulness of the proposed structural topology optimization method.

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary-based moving superimposed finite element method (s-version FEM or S-FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S-FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co-ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S-FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S-FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well-recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S-FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.

Topology optimization has been successfully used in many industries, especially those engaged in the design and manufacturing of mechanical devices, but numerical problems are often encountered, such as grayscale representations of obtained composites. A type of structural optimization method using the level set theory for boundary expressions has been proposed, in which the outlines of target structures are implicitly represented using the level set function, and optimal configurations are obtained by updating this function based on the shape sensitivities. Level set-based methods typically have a drawback, however, in that topological changes that increase the number of holes in the material domain are not allowed. To overcome the above numerical and topological problems, this paper proposes a new topology optimization method incorporating level set boundary expressions based on the concept of the phase field method, which we apply to a minimum mean compliance problem. First, a structural optimization problem is formulated based on a boundary expression, using the level set function. Next, a time evolutionary equation for updating the level set function is formulated based on the concept of the phase field method, and the minimum mean compliance problem is formulated using a level set boundary expression. An optimization algorithm for the topology optimization incorporating the level set boundary expression based on the concept of the phase field method is then derived. Several examples are provided to confirm the usefulness of the proposed structural topology optimization method.

In bone-remodeling studies, it is believed that the morphology of bone is affected by its internal mechanical loads. From the 1970s, high computing power enabled quantitative studies in the simulation of bone remodeling or bone adaptation. Among them, Huiskes et al. (1987, "Adaptive Bone Remodeling Theory Applied to Prosthetic Design Analysis," J. Biomech. Eng., 20, pp. 1135-1150) proposed a strain energy density based approach to bone remodeling and used the apparent density for the characterization of internal bone morphology. The fundamental idea was that bone density would increase when strain (or strain energy density) is higher than a certain value and bone resorption would occur when the strain (or strain energy density) quantities are lower than the threshold. Several advanced algorithms were developed based on these studies in an attempt to more accurately simulate physiological bone-remodeling processes. As another approach, topology optimization originally devised in structural optimization has been also used in the computational simulation of the bone-remodeling process. The topology optimization method systematically and iteratively distributes material in a design domain, determining an optimal structure that minimizes an objective function. In this paper, we compared two seemingly different approaches in different fields-the strain energy density based bone-remodeling algorithm (biomechanical approach) and the compliance based structural topology optimization method (mechanical approach)-in terms of mathematical formulations, numerical difficulties, and behavior of their numerical solutions. Two numerical case studies were conducted to demonstrate their similarity and difference, and then the solution convergences were discussed quantitatively.

The trabecular bone adapts its form to mechanical loads and is able to form lightweight but very stiff structures. In this sense, it is a problem (for the Nature) similar to the structural optimization, especially the topology optimization. The structural optimization system based on shape modification using shape derivative [1] will be presented. Structural evolution during the structural optimization procedure is based on the adaptive remodeling of the trabecular bone and is independent of domain selection. In general the problem of stiffest design (compliance minimization) has no solution. If the volume of the object is increasing, the compliance is decreasing. Thus, in the standard approach to the energy based topology optimization the additional constraint has to be added. Usually the volume of the material is limited. But with such an assumption the optimization procedure does not include any criteria for stress. In case of the biomimetic approach presented here, the role of the additional constraint plays the strain energy density on the structural surface (which is between two assumed levels forming the insensitivity zone) and the volume or structural mass results from the optimization procedure. The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a heuristic algorithm. The method is enhanced to handle the problem of structural optimization under multiple loads [2]. The new approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface.

This paper presents a method for assisting the optimal selection of topologies for the minimum-weight design of continuum structures subject to stress constraints by using the performance index (PI). A performance index is developed for evaluating the efficiency of structural topologies based on the scaling design approach. This performance index is incorporated in the evolutionary structural optimization (ESO) method to monitor the optimization process when gradually removing inefficient material from the structure. The optimal topology can be identified from the performance index history. Various structures with stress and height constraints are investigated by using this performance index, which is also employed to compare the efficiency of structural topologies generated by different optimization methods. It is shown that the proposed performance index is capable of measuring the efficiency of structural topologies obtained by various structural optimization methods and is a valuable design tool for engineers in selecting optimal topologies in structural design.

Discrete design variables are encountered in most practical design applications. An overview of the optimization methods that can treat such design variables in their solution process is presented. Many times selection of an appropriate method depends on type of discrete variables and problem functions. Therefore, discrete design variable problems are classified into six different types. Methods suitable for each problem type are identified. Some recent applications of discrete variable optimization methods are described. Methods for structural optimization with available sections are discussed.

Most research studies on structural optimum design have focused on single-objective optimization of deterministic structures, while little study has been carried out to address multi-objective optimization of random structures. Statistical parameters and redundancy allocation problems should be considered in structural optimization. In order to address these problems, this paper presents a hybrid method for structural system reliability-based design optimization (SRBDO) and applies it to trusses. The hybrid method integrates the concepts of the finite element method, radial basis function (RBF) neural networks, and genetic algorithms. The finite element method was used to compute structural responses under random loads. The RBF neural networks were employed to approximate structural responses for the purpose of replacing the structural limit state functions. The system reliabilities were calculated by Monte Carlo simulation method together with the trained RBF neural networks. The optimal parameters were obtained by genetic algorithms, where the system reliabilities were converted into penalty functions in order to address the constrained optimization. The hybrid method applied to trusses was demonstrated by two examples which were a typical 10-bar truss and a steel truss girder structure. Detailed discussions and parameter analysis for the failure sequences such as web-bucking failure and beam-bending failure in the SRBDO were given. This hybrid method provides a new idea for SRBDO of trusses. Copyright © 2015 John Wiley & Sons, Ltd.

Further improvement of engines' efficiency is becoming increasingly challenging, especially for small-scale gas turbines. A promising approach to achieve this goal is to combine engine's structural optimization with a parametric optimization of its working process. The array of applications requiring small-scale gas turbines with the increasing values of working process parameters is constantly expanding. Even turbofan cores of full-size airliners are essentially becoming small-scale with ever-increasing bypass ratios. Thus, updating the mathematical models used for the optimization of engine's working process to better describe impact of sizing factor on the engine performance is becoming a pressing matter. This paper introduces the method of joint structural and parametric optimization of small-scale jet engines and describes the updates made to the mathematical models of engine components. Verification of the developed software code is done by comparing the optimization results with the commercially available turbofan DGEN 380 developed by Price Induction.

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

Effective component design synthesis using structural and shape optimization methods

Most structural optimization methods are based on mathematical algorithms that require substantial gradient information. The selection of the starting values is also important to ensure that the algorithm converges to the global optimum. This paper describes a new structural configuration optimization method based on the harmony search (HS) meta-heuristic algorithm. The HS algorithm does not require initial values and uses a random search instead of a gradient search, so derivative information is unnecessary. A benchmark truss example is presented to demonstrate the effectiveness and robustness of the proposed approach compared to other optimization methods. Results reveal that the proposed approach is capable of solving configuration optimization problems, and may yield better solutions than those obtained using earlier methods.

This article introduces the element-propagating method to structural shape and topology optimization. Structural optimization based on the conventional level-set method needs to solve several partial differential equations. By the insertion and deletion of basic material elements around the geometric boundary, the element-propagating method can avoid solving the partial differential equations and realize the dynamic updating of the material region. This approach also places no restrictions on the signed distance function and the Courant–Friedrichs–Lewy condition for numerical stability. At the same time, in order to suppress the dependence on the design initialization for the 2D structural optimization problem, the strain energy density is taken as a criterion to generate new holes in the material region. The coupled algorithm of the element-propagating method and the method for generating new holes makes the structural optimization more robust. Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the conventional level-set method for structural topology optimization.

Evaluation and comparison of geometry representation methods for structural topology optimization

Evaluation and comparison of geometry representation methods for structural topology optimization