Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

The paper presents a gradient‐based topology optimization formulation that allows to solve acoustic–structure (vibro‐acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface‐coupling integrals, however, such a formulation does not allow for free material re‐distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u /p‐formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two‐dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

A nodal variable method of structural topology optimization based on Shepard interpolant

A method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiresolution topology optimization using isogeometric analysis

The paper introduces a novel multiresolution scheme to topology optimization in the framework of the isogeometric analysis. A new variable parameter space is added to implement multiresolution topology optimization based on the Solid Isotropic Material with Penalization approach. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B‐spline basis functions, which are easily obtained using k‐refinement strategy in the isogeometric analysis. While the nonuniform rational B‐spline basis functions are used to exactly describe geometric domains and approximate unknown solutions in finite element analysis. By applying a refined sensitivity filter, optimized designs include highly discrete solutions in terms of solid and void materials without using any black and white projection filters. The Method of Moving Asymptotes is used to solve the optimization problem. Various benchmark test problems including plane stress, compliant mechanism inverter, and 2‐dimensional heat conduction are examined to demonstrate the effectiveness and robustness of the present method.     

Slope constrained topology optimization

The problem of minimum compliance topology optimization of an elastic continuum is considered. A general continuous density–energy relation is assumed, including variable thickness sheet models and artificial power laws. To ensure existence of solutions, the design set is restricted by enforcing pointwise bounds on the density slopes. A finite element discretization procedure is described, and a proof of convergence of finite element solutions to exact solutions is given, as well as numerical examples obtained by a continuation/SLP (sequential linear programming) method. The convergence proof implies that checkerboard patterns and other numerical anomalies will not be present, or at least, that they can be made arbitrarily weak. © 1998 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     

离散变量桁架结构拓扑优化设计的混合算法

将相对差商法和混沌优化结合起来,形成求解离散变量桁架结构拓扑优化设计的混合算法。利用相对差商法可以对离散变量快速寻优的特点,及混沌变量的全局遍历性,可以有效地跳出局部最优解,达到拓扑优化全局寻优的目的。通过采用和准最优解的对比及几何稳定性的判断等辅助性技术,降低了重分析次数。同时,高效的重分析方法的结合,提高了求解的效率,也避免了拓扑优化问题中求解的一些困难。算例表明,该算法对于离散变量的拓扑优化设计问题是快速有效的。     

用于投射式电子束曝光系统的磁浸没透镜的优化设计

以磁浸没透镜的电流密度值的大小为优化参量,以成像系统的轴上像散为目标函数,利用单纯形优化法对用于投射式电子束曝光系统的新型磁浸没透镜进行了优化设计.结果表明,目标函数由未经优化前的358.187nm降低到50.693nm.同时单纯形优化法不需要已知目标函数与优化参量之间的解析关系式及求导,因此作为电子光学系统的最优化设计具有明显的优越性.     

An asymptotically concentrated method for structural topology optimization based on the SIMLF interpolation

In this work, an asymptotically concentrated topology optimization method based on the solid isotropic material with logistic function interpolation is proposed. The asymptotically concentrated method is introduced into the process of optimization cycle after updating the design variables. At the same time, with the use of the solid isotropic material with logistic function interpolation, all the candidate densities are reasonably polarized, relying on the characteristic of the interpolation curve itself. The asymptotically concentrated method can effectively suppress the generation of intermediate density and speed up the process of updating the design variables, hence improving the optimization efficiency. Moreover, the above polarization can weaken the influence of low‐related‐density elements and enhance the influence of high‐related‐density elements. For the single‐material topology optimization problem, gray‐scale elements can be effectively eliminated, and clear boundary and smaller compliance can be obtained by this method. For the multimaterial topology optimization problem, minimum compliance with high efficiency can be achieved by this method. The proposed method mainly includes the following advantages: concentrated density variables, reasonable interpolation, high computational efficiency, and good topological results.     

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

Simultaneous shape optimization of thin‐walled curved shell structures and involved hole boundaries is studied in this paper. A novel bispace parameterization method is proposed for the first time to define global and local shape design variables both in the Cartesian coordinate system and the intrinsic coordinate system. This method has the advantage of achieving a simultaneous optimization of the global shape of the shell surface and the local shape of the openings attached automatically on the former. Inherent problems, for example, the effective parameterization of shape design variables, mapping operation between two spaces, and sensitivity analysis with respect to both kinds of design variables are highlighted. A design procedure is given to show how both kinds of design variables are managed together and how the whole design flowchart is carried out with relevant formulations. Numerical examples are presented and the effects of both kinds of design variables upon the optimal solutions are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

Shape optimization with topological changes and parametric control

Recent advances in shape optimization rely on free-form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher-dimensional function that is constructed using B-splines (to allow free-form deformations), and parameterized primitives combined with R-functions (to support desired parametric changes). Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free-form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient-based optimization methods. Furthermore, our implementation does not require any error-prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Isogeometric topology optimization for continuum structures using density distribution function

This paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1)  Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high-order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.     

Achieving minimum length scale in topology optimization using nodal design variables and projection functions

A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non‐linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user‐defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.     

Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations

We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of coercive elliptic partial differential equations with affine (input) parameter dependence. The critical ingredients are: reduced-basis approximation to effect significant reduction in state-space dimensionality; a posteriori error bounds to provide rigorous error estimation and control; “offline/online” computational decompositions to permit rapid evaluation of output bounds, output bound gradients, and output bound Hessians in the limit of many queries; and reformulation of the approximate optimization statement to ensure (true) feasibility and control of suboptimality. To illustrate the method we consider the design of a three-dimensional thermal fin: Given volume and power objective-function weights, and root temperature “not-to-exceed” limits, the optimal geometry and heat transfer coefficient can be determined—with guaranteed feasibility—in only 2.3 seconds on a 500 MHz Pentium machine; note the latter includes only the online component of the calculations. Our method permits not only interactive optimal design at conception and manufacturing, but also real-time reliable adaptive optimal design in operation.     

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.     

Sensitivity analysis and shape optimization of axisymmetric structures

An efficient method is developed for sensitivity analysis in shape optimization of axisymmetric structures. The technique of isoparametric mapping is used to generate the finite element mesh from a small set of master elements and master nodes. Co-ordinates of selected master nodes are used as design variables. Shape function values of master elements at derived finite element nodes obtained during the isoparametric mapping process are utilized to calculate the gradients of weight and response of the structures with respect to the design variables. Analytic formulations of the gradients are developed for sensitivity analysis of axisymmetric structures. An optimization procedure using a sequential linear programming method is applied to effectively utilize the calculated gradients. Numerical examples of optimum design of disks subject to thermo-mechanical loadings are presented.     

考虑结构稳定性的变密度拓扑优化方法

将稳定性问题引入传统变密度法中,可实现包含稳定性约束的平面模型结构拓扑优化。以单元相对密度为设计变量,结构柔度最小为目标函数,结构体积和失稳载荷因子为约束条件建立优化问题数学模型,提出了一种考虑结构稳定性的变密度拓扑优化方法。通过分析结构柔度、体积、失稳载荷因子对设计变量的灵敏度,并基于拉格朗日乘子法和Kuhn-Tucker条件,推导了优化问题的迭代准则。同时,利用基于约束条件的泰勒展开式求解优化准则中的拉格朗日乘子。通过推导平面四节点四边形单元几何刚度矩阵的显式表达式,得到了优化准则中的几何应变能。最后,通过算例对提出的方法进行了验证,并与不考虑稳定性的传统变密度拓扑优化方法进行对比,结果表明该方法能显著提高拓扑优化结果的稳定性。研究结果对细长受压结构的优化设计有重要指导意义,对结构的稳定性设计有一定参考价值。     

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.     

Cellular topology optimization on differentiable Voronoi diagrams

Cellular structures manifest their outstanding mechanical properties in many biological systems. One key challenge for designing and optimizing these geometrically complicated structures lies in devising an effective geometric representation to characterize the system's spatially varying cellular evolution driven by objective sensitivities. A conventional discrete cellular structure, for example, a Voronoi diagram, whose representation relies on discrete Voronoi cells and faces, lacks its differentiability to facilitate large-scale, gradient-based topology optimizations. We propose a topology optimization algorithm based on a differentiable and generalized Voronoi representation that can evolve the cellular structure as a continuous field. The central piece of our method is a hybrid particle-grid representation to encode the previously discrete Voronoi diagram into a continuous density field defined in a Euclidean space. Based on this differentiable representation, we further extend it to tackle anisotropic cells, free boundaries, and functionally-graded cellular structures. Our differentiable Voronoi diagram enables the integration of an effective cellular representation into the state-of-the-art topology optimization pipelines, which defines a novel design space for cellular structures to explore design options effectively that were impractical for previous approaches. We showcase the efficacy of our approach by optimizing cellular structures with up to thousands of anisotropic cells, including femur bone and Odonata wing.