Constrained isogeometric design optimization of lattice structures on curved surfaces: computation of design velocity field

This paper presents an immersed boundary approach for level set topology optimization considering stress constraints. A constraint agglomeration technique is used to combine the local stress constraints into one global constraint. The structural response is predicted by the eXtended Finite Element Method. A Heaviside enrichment strategy is used to model strong and weak discontinuities with great ease of implementation. This work focuses on low-order finite elements, which given their simplicity are the most popular choice of interpolation for topology optimization problems. The predicted stresses strongly depend on the intersection configuration of the elements and are prone to significant errors. Robust computation of stresses, regardless of the interface position, is essential for reliable stress constraint prediction and sensitivities. This study adopts a recently proposed fictitious domain approach for penalization of displacement gradients across element faces surrounding the material interface. In addition, a novel XFEM informed stabilization scheme is proposed for robust computation of stresses. Through numerical studies the penalized spatial gradients combined with the stabilization scheme is shown to improve prediction of stresses along the material interface. The proposed approach is applied to the benchmark topology optimization problem of an L-shaped beam in two and three dimensions using material-void and material-material problem setups. Linear and hyperelastic materials are considered. The stress constraints are shown to be efficient in eliminating regions with high stress concentration in all scenarios considered.     

Multiparametric shape optimal design of disks

The external boundary of a structure described by a set of design parameters undergoes shape modification. Arbitrary stress, strain and displacement functionals are defined within the domain of the structure and its first- and second-order sensitivities with respect to varying structural shape are discussed. The optimal shape design problem is then formulated and solved using the first- and second-order sensitivity information. The iterative analysis-redesign algorithm is formulated using the finite element method. Some illustrative examples are included.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

Single scale wavelet approximations in layout optimization

The standard structural layout optimization problem in 2D elasticity is solved using a wavelet based discretization of the displacement field and of the spatial distribution of material. A fictitious domain approach is used to embed the original design domain within a simpler domain of regular geometry. A Galerkin method is used to derive discretized equations, which are solved iteratively using a preconditioned conjugate gradient algorithm. A special preconditioner is derived for this purpose. The method is shown to converge at rates that are essentially independent of discretization size, an advantage over standard finite element methods, whose convergence rate decays as the mesh is refined. This new approach may replace finite element methods in very large scale problems, where a very fine resolution of the shape is needed. The derivation and examples focus on 2D-problems but extensions to 3D should involve only few changes in the essential features of the procedure.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

An efficient shape optimization technique of a two-dimensional body based on the boundary element method

An efficient approximation method to determine the optimum shape of the minimum weight of a body subjected to stress and displacement constraints is suggested by using the boundary element method. The objective function of weight is approximated to an expansion of a second-order Taylor series and the stress and displacement constraints to expansions of a first-order Taylor series, based on the boundary element sensitivity analysis at the current design point. The approximated subproblem is then solved by a linear complementary pivot method. Design variable reduction techniques of isoparametric interpolation and trigonometric series interpolation for the design boundary shape are also adopted for reducing the degrees of freedom of the design problems.     

E(FG)<Superscript>2</Superscript>: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.     

Shape design of elastostatic structures based on local perturbation analysis

This paper describes a non-gradient formulation for solving shape optimal design problems involving structures in plane stress or having an axially symmetric geometry. The minimization of the maximum von Mises stress value at a traction free boundary poses a non-linear optimization problem in which the design variables do not appear explicitly in the formulation. The most commonly used approach is to apply a standard non-linear programming technique. There exists in this field no universally accepted solution method. The major difficulty of shape optimization in connection with FEM is to perform an accurate and efficient sensitivity analysis. The perturbation analysis introduced here takes advantage of the character of the problem. It is based on methods from the theory of notches. The results are applied to an FE-model of the structural component. The iterative method with such a direction of search works efficiently even for a large number of design variables as shown by Schnack (1977b, 1978, 1979, 1980, 1983 and 1985). Using a dynamic programming formulation (see also Schnack and Spörl 1986), the existence of a solution for the shape optimal problem will be discussed. Examples of applications to structural components from mechanical engineering are presented to demonstrate the power of this approach.     

Continuum shape sensitivity analysis of a mixed-mode fracture in functionally graded materials

This paper presents two new methods for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic functionally graded material. These methods involve the material derivative concept from continuum mechanics, domain integral representation of interaction integrals, known as the M-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the sensitivity of stress–intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. Three numerical examples are presented to calculate the first-order derivative of the stress–intensity factors. The results show that first-order sensitivities of stress intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Compatibility Condition in Theory of Solid Mechanics (Elasticity,Structures, and Design Optimization)

The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element structural analysis. This has lead to the completion of the “method of force” with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in the finite element analysis. The dual integrated force method (IFMD) with displacement as the primary unknown had been formulated. Both the IFM and IFMD produce identical responses. The IFMD can utilize the equation solver of the traditional stiffness method. The variational derivation of the CBMF produced the existing sets of elasticity equations along with the new boundary compatibility conditions, which were missed since the time of Saint-Venant, who formulated the field equations about 1860. The CBMF, which can be used to solve stress, displacement, and mixed boundary value problems, has eliminated the restriction of the classical method that was applicable only to stress boundary value problem. The IFM in structures produced high-fidelity response even with a modest finite element model. Because structural design is stress driven, the IFM has influenced it considerably. A fully utilized design method for strength and stiffness limitation was developed via the IFM analysis tool. The method has identified the singularity condition in structural optimization and furnished a strategy that alleviated the limitation and reduced substantially the computation time to reach the optimum solution. The CBMF and IFM tensorial approaches are robust formulations because both methods simultaneously emphasize the equilibrium equation and the compatibility condition. The vectorial displacement method emphasized the equilibrium, while the compatibility condition became the basis of the scalar stress-function approach. The tensorial approach can be transformed to obtain the vector and the scalar methods, but the reverse course cannot be followed. The tensorial approach outperformed other methods as expected. This paper introduces the new concepts in elasticity, in finite element analysis, and in design optimization with numerical illustrations.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

2-D and 3-D shape optimization using mesh velocities to integrate analytical sensitivities with associative CAD

The design sensitivities generated with the mesh velocity method, used by the authors, are compared with those obtained by the boundary layer and boundary displacement methods. The effect of adaptive mesh refinement and error control on the quality of the velocity fields is discussed, as well as their ability to yield accurate first-order predictions of constraint values. Two numerical shape optimization examples of a 2D and a 3D component are presented. These examples are used to illustrate the benefits of integrating analytical methods of design sensitivity analysis with parametric capabilities supported by state-of-the-art CAD systems.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Design of absorbing material distribution for sound barrier using topology optimization

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0–1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton–Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.     

Boundary formulations for sensitivities of three-dimensional stress invariants

The direct, singular, boundary element analysis (BEA) formulation has been shown to provide a basis for a computationally efficient and accurate shape structural design sensitivity analysis (DSA) approach for three-dimensional solid objects. Within the boundary element analysis context, the theoretical formulation for sensitivities of important stress-related quantities including principal and deviatoric stresses, von Mises, maximum shear, and other stress invariants are presented, both for the surface as well as the interior of a continuum structure. Numerical results are given to demonstrate the accuracy of this approach.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

层次式直接边界元计算VLSI三维互连电容

文中将Ａｐｐｅｌ处理多体问题的层次式算法思想实现于直接边界元法,用以计算ＶＬＳＩ三维互连寄生电容。直接边界积分方程同时含有边界上的电势与法向电场强度,能比间接边界元法更方便地处理多介质及有限介质结构,直接边界元法的层次式计算涉及对三种边界（强加边界、自然边界与介质交界面）及两种积分核（１／ｒ与１／ｒ＾３）的处理,显著区别于基于间接边界元法、仅处理强加边界与一种分核的层次式算法。文中以边界元的层次划     

求解应力强度因子的样条虚边界元交替法

为求解平面裂纹问题的应力强度因子,提出基于Muskhelishvili基本解和样条虚边界元法的样条虚边界元交替法.该方法将平面内带裂纹有限域问题分解成带裂纹无限域问题与不带裂纹有限域问题的叠加.带裂纹无限域问题利用Muskhelishvili基本解法直接得出,不带裂纹有限域问题采用样条虚边界元法求解.利用该方法对复合型中心裂纹方板和I型偏心裂纹矩形板进行分析.数值结果表明该方法精度高且适用性强.     

A non-parametric free-form optimization method for shell structures

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures.     

Topology optimization in crashworthiness design

Topology optimization has developed rapidly, primarily with application on linear elastic structures subjected to static loadcases. In its basic form, an approximated optimization problem is formulated using analytical or semi-analytical methods to perform the sensitivity analysis. When an explicit finite element method is used to solve contact–impact problems, the sensitivities cannot easily be found. Hence, the engineer is forced to use numerical derivatives or other approaches. Since each finite element simulation of an impact problem may take days of computing time, the sensitivity-based methods are not a useful approach. Therefore, two alternative formulations for topology optimization are investigated in this work. The fundamental approach is to remove elements or, alternatively, change the element thicknesses based on the internal energy density distribution in the model. There is no automatic shift between the two methods within the existing algorithm. Within this formulation, it is possible to treat nonlinear effects, e.g., contact–impact and plasticity. Since no sensitivities are used, the updated design might be a step in the wrong direction for some finite elements. The load paths within the model will change if elements are removed or the element thicknesses are altered. Therefore, care should be taken with this procedure so that small steps are used, i.e., the change of the model should not be too large between two successive iterations and, therefore, the design parameters should not be altered too much. It is shown in this paper that the proposed method for topology optimization of a nonlinear problem gives similar result as a standard topology optimization procedures for the linear elastic case. Furthermore, the proposed procedures allow for topology optimization of nonlinear problems. The major restriction of the method is that responses in the optimization formulation must be coupled to the thickness updating procedure, e.g., constraint on a nodal displacement, acceleration level that is allowed.