Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance–volume product

In this paper the concept of extended optimality, or hyperoptimality, is adopted. By following this idea, a new compliance–volume product is suggested as objective. The volume appearing in the product is also raised to the power of a new design parameter which can be set to different values. In such manner design concepts with different volume fractions can be generated by using the approach of extended optimality. Both manufacturing constraints and unilateral contact constraints are included in the proposed method. The manufacturing constraints are implemented by adjusting the move limits such that the draw directions are satisfied. Both one draw direction as well as split draw constraints are considered. The contact conditions are modeled by the augmented Lagrangian approach such that the Jacobian in the Newton algorithm as well as in the adjoint equation becomes symmetric. The design parametrization is done by the SIMP model and Sigmund’s filter is utilized when the sensitivities are calculated. The proposed method is very robust and efficient. This is demonstrated by solving problems in both 2D and 3D. The numerical results are also compared to solutions obtained by performing compliance optimization with a constraint on the volume fraction.     

Contact shape optimization: a bilevel programming approach

We consider the problem of shape optimization of nonlinear elastic solids in contact. The equilibrium of the solid is defined by a constrained minimization problem, where the body energy functional is the objective and the constraints impose the nonpenetration condition. Then the optimization problem can be formulated in terms of a bilevel mathematical program. We describe new optimality conditions for bilevel programming and construct an algorithm to solve these conditions based on Herskovits’ feasible direction interior point method. With this approach we simultaneously carry out shape optimization and nonlinear contact analysis. That is, the present method is a “one shot” technique. We describe some numerical examples solved in a very efficient way. Received July 27, 1999     

Stable computation of Gaussian interpolants—a heuristic approach

Approximations play an important role in optimization to reduce the number of expensive analysis runs. In particular, metamodels are one way to reduce CPU time for multidisciplinary optimization and are sometimes the enabler for huge optimization projects. One type of approximation is interpolation and radial basis functions (RBFs) are an example for it. In general, a large number of radial basis functions have the attractive property that—independently from the arrangement of sampling points—the existence of a unique solution can be guaranteed for the linear equation system that is solved to determine the coefficients. However, this doesn’t mean that the handling is uncritical, as can be seen for radial basis functions of Gaussian type (GRBFs). In this case ill-conditioning and Runge-type oscillations spoil the tuning of the interpolant’s shape parameter and make its general application as a metamodel impossible. We introduce a heuristic approach to modify the GRBFs in a way that allows the shape parameter to be optimized within a much larger range before ill-conditioning appears. It seems that also the appearance of the Runge-type oscillations are solved by this approach.     

Multicriteria optimization based on neural network approximation of decision maker’s utility function

A direct adaptive method of multicriteria optimization based on neural network approximation of a decision maker’s utility function is introduced. Efficiency of method is analyzed via two- and three- criterion test problems solving. Efficiency testing results are shown. The method implementation as an optimization module for PRADIS (Program for Analysis of Dynamic Systems) is described. Practical application of the method for solving two-criterion optimization problem of combustion engine mechanical subsystem is carried out.     

Study of variational inequality and equality formulations for elastostatic frictional contact problems

Summary An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.     

Topology optimization of structures with geometrical nonlinearities

In this paper, the stiffness optimization problem is investigated for structures with geometrical nonlinearities. The mean compliance of the structure is chosen as the objective function and its sensitivities are derived using the adjoint method. The optimal problem is formulated using a microstructure-based design domain method and is solved iteratively by a sequential convex approximation method. Three numerical examples are presented to show the applications of the proposed method and the results demonstrate that the geometrically nonlinear finite element analysis is indispensable to the optimization process of large displacement structures.     

Efficient use of iterative solvers in nested topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of the analysis equations. In this study, it is suggested to reduce this computational cost by using an approximation to the solution of the analysis problem, generated by a Krylov subspace iterative solver. By choosing convergence criteria for the iterative solver that are strongly related to the optimization objective and to the design sensitivities, it is possible to terminate the iterative solution of the nested equations earlier compared to traditional convergence measures. The approximation is computationally shown to be sufficiently accurate for the purpose of optimization though the nested equation system is not necessarily solved accurately. The approach is tested on several large-scale topology optimization problems, including minimum compliance problems and compliant mechanism design problems. The optimized designs are practically identical while the time spent on the analysis is reduced significantly.     

Rowwise aggregation of variables in the dynamic programming algorithm for image processing

There is a wide class of image processing problems that can be formulated as optimization problems (in particular, as (min, +) labeling problems) on lattice adjacency graphs that describe the adjacency of variables. In the general case, such problems are known to be NP-complete. However, if the adjacency graph is acyclic, they can easily be solved using dynamic programming. There are many optimization techniques that use a partitioning or an approximation of a lattice graph by a set of acyclic graphs; however, some (min, +) labeling problems cannot be solved using such techniques. In this paper, an optimization technique based on a rowwise combination of variables (rather than on decomposition into acyclic graphs) is proposed that also enables one to use dynamic programming. Andrei V. Kopylov. Born 1970. Received master’s degree from Tula State University in 1993. Received candidate’s degree in engineering in 1997. Assistant professor of the Chair of Automation and Remote Control at Tula State University. Scientific interests: image and signal analysis, image matching, smoothing signals and images while retaining edges. In 1997 awarded MAIK “Nauka/Interperiodica” prize for the best series of publications in scientific journals of the Russian Academy of Sciences. Author of more than 50 papers; member of the Russian Division of the International Association of Pattern Recognition.     

Multimodal identification and tracking in smart environments

We present a model for unconstrained and unobtrusive identification and tracking of people in smart environments and answering queries about their whereabouts. Our model supports biometric recognition based upon multiple modalities such as face, gait, and voice in a uniform manner. The key technical idea underlying our approach is to abstract a smart environment by a state transition system in which each state records a set of individuals who are present in various zones of the environment. Since biometric recognition is inexact, state information is inherently probabilistic in nature. An event abstracts a biometric recognition step, and the transition function abstracts the reasoning necessary to effect state transitions. In this manner, we are able to integrate different biometric modalities uniformly and also different criteria for state transitions. Fusion of biometric modalities is also supported by our model. We define performance metrics for a smart environment in terms of the concepts of ‘precision’ and ‘recall’. We have developed a prototype implementation of our proposed concepts and provide experimental results in this paper. Our conclusion is that the state transition model is an effective abstraction of a smart environment and serves as a good basis for developing practical systems.     

A shape optimization study for tool design in resistance welding

The purpose of this study is to apply shape optimization tools for design of resistance welding electrodes. The numerical simulation of the welding process has been performed by a simplified FEM model implemented in COMSOL. The design process is formulated as an optimization problem where the objective is to prolong the life-time of the electrodes. Welding parameters like current, time and electrode shape parameters are selected to be the design variables while constraints are chosen to ensure a high quality of the welding. Surrogate models based on a Kriging approximation has been used in order to simplify the calculation of shape sensitivities and to generate a generic tool that can be interfaced with other simulation tools. An example numerical study shows the potential of applying optimal design techniques in this area. Part of this work was presented at WCSMO7 in Seoul Korea, May 21–25, 2007, in the paper titled ‘Some optimization aspects of resistance welding’ (CD-ROM, pp 2687–2695).     

Topology optimization of flow domains using the lattice Boltzmann method

We consider the optimal design of two- (2D) and three-dimensional (3D) flow domains using the lattice Boltzmann method (LBM) as an approximation of Navier-Stokes (NS) flows. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. NS flows are traditionally approximated by finite element and finite volume methods. These schemes, while well established as high-fidelity simulation tools using body-fitted meshes, are effected in their accuracy and robustness when regular meshes with zero-velocity constraints along the surface and in the interior of obstacles are used, as is common in topology optimization. Therefore, we study the potential of the LBM for approximating low Mach number incompressible viscous flows for topology optimization. In the LBM the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material-based topology optimization. In addition, this non-traditional discretization method features parallel scalability and allows for high-resolution, regular fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the LBM for the optimal design of fluid domains, making the LBM an interesting alternative to NS solvers for topology optimization problems. The potential of our topology optimization approach will be illustrated by 2D and 3D numerical examples.     

The Painlevé paradox studied at a 3D slender rod

This paper aims at investigating the dynamical behaviors of a 3D rod moving on a rough surface with so-called Painlevé paradox. The condition for the occurrence of the Painlevé paradox in the rod is studied according to the theoretical results obtained from LCP’s method for spatial multibody systems. Numerical results obtained by inserting a compliant contact model into the rigid body model present a support for the assumption that a tangential impact is related to the spatial paradoxical situations. Furthermore, the tangential impact is analyzed by using the Darboux–Keller’s shock dynamics and are found with the same properties as the one in the planar rod: A tangential stick appears at the contact point during the impulsive process. With the help of the Stronge’s coefficient, an impact rule is developed to describe the dynamical behaviors of the 3D rod with paradoxical situations. Comparisons between numerical results obtained from Darboux’s model and the ones obtained from the compliant contact model are carried out and show well agreements.     

A Tensor Approximation Approach to Dimensionality Reduction

Dimensionality reduction has recently been extensively studied for computer vision applications. We present a novel multilinear algebra based approach to reduced dimensionality representation of multidimensional data, such as image ensembles, video sequences and volume data. Before reducing the dimensionality we do not convert it into a vector as is done by traditional dimensionality reduction techniques like PCA. Our approach works directly on the multidimensional form of the data (matrix in 2D and tensor in higher dimensions) to yield what we call a Datum-as-Is representation. This helps exploit spatio-temporal redundancies with less information loss than image-as-vector methods. An efficient rank-R tensor approximation algorithm is presented to approximate higher-order tensors. We show that rank-R tensor approximation using Datum-as-Is representation generalizes many existing approaches that use image-as-matrix representation, such as generalized low rank approximation of matrices (GLRAM) (Ye, Y. in Mach. Learn. 61:167–191, 2005), rank-one decomposition of matrices (RODM) (Shashua, A., Levin, A. in CVPR’01: Proceedings of the 2001 IEEE computer society conference on computer vision and pattern recognition, p. 42, 2001) and rank-one decomposition of tensors (RODT) (Wang, H., Ahuja, N. in ICPR ’04: ICPR ’04: Proceedings of the 17th international conference on pattern recognition (ICPR’04), vol. 1, pp. 44–47, 2004). Our approach yields the most compact data representation among all known image-as-matrix methods. In addition, we propose another rank-R tensor approximation algorithm based on slice projection of third-order tensors, which needs fewer iterations for convergence for the important special case of 2D image ensembles, e.g., video. We evaluated the performance of our approach vs. other approaches on a number of datasets with the following two main results. First, for a fixed compression ratio, the proposed algorithm yields the best representation of image ensembles visually as well as in the least squares sense. Second, proposed representation gives the best performance for object classification. A shorter version of this paper was published at IEEE CVPR 2005 (Wang and Ahuja 2005).     

Primal-dual Newton Methods in Structural Optimization

We consider the numerical solution of optimization problems for systems of partial differential equations with constraints on the state and design variables as they arise in the optimal design of the shape and the topology of continuum mechanical structures. After discretization the resulting nonlinear programming problems are solved by an “all-at-once” approach featuring the numerical solution of the state equations as an integral part of the optimization routine. In particular, we focus on primal-dual Newton methods combined with interior-point techniques for an appropriate handling of the inequality constraints. Special emphasis is given on the efficient solution of the primal-dual system that results from the application of Newton’s method to the Karush–Kuhn–Tucker conditions where we take advantage of the special block structure of the primal-dual Hessian. Applications include structural optimization of microcellular biomorphic ceramics by homogenization modeling, the shape optimization of electrorheological devices, and the topology optimization of high power electromotors.Dedicated to Peter Deuflhard on the occasion of his 60th birthdayThe first and the third author have been supported by the DFG within the Collaborative Research Center SFB 438 and within the Priority Program SPP 1095 under the Grants Ho 877/5-1 and Ho 877/5-2. The first author acknowledges further support by the BMBF under Grant 03HOM3A1     

On shape sensitivities with heaviside-enriched XFEM

This paper investigates the behavior of shape sensitivities within the context of the eXtended Finite Element Method (XFEM) using a Heaviside enrichment strategy, wherein the shape derivative is computed by the adjoint method. The Heaviside function is discontinuous by construction. This feature of the enrichment function presents advantages as well as challenges in the computation of shape sensitivities, both of which are discussed in detail in this paper. Using continuum and discrete approaches, we present the derivation of analytical shape sensitivities with respect to the design variables which define the design geometry. We propose a robust semi-analytical approach to computing the shape sensitivities, which provides great ease of implementation as compared to fully analytical approaches. The behavior of the XFEM-based shape sensitivities is analyzed using linear heat diffusion examples in 2D, and an incompressible fluid flow example in 3D. We compare XFEM-based shape sensitivities against shape sensitivities obtained through the classical approach of using a body-fitted mesh. It is found that the former are not as smooth as those obtained using a comparable body-fitted mesh. This discrepancy is shown to be an outcome of the discretization error of the design geometry on a background mesh and is not a consequence of the approach by which the XFEM-based shape sensitivities are computed.     

An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems

This paper presents an approach to shape and topology optimization of fluid-structure interaction (FSI) problems at steady state. The overall approach builds on an immersed boundary method that couples a Lagrangian formulation of the structure to an Eulerian fluid model, discretized on a deforming mesh. The geometry of the fluid-structure boundary is manipulated by varying the nodal parameters of a discretized level set field. This approach allows for topological changes of the fluid-structure interface, but free-floating volumes of solid material can emerge in the course of the optimization process. The free-floating volumes are tracked and modeled as fluid in the FSI analysis. To sense the isolated solid volumes, an indicator field described by linear, isotropic diffusion is computed prior to analyzing the FSI response of a design. The fluid is modeled with the incompressible Navier-Stokes equations, and the structure is assumed linear elastic. The FSI model is discretized by an extended finite element method, and the fluid-structure coupling conditions are enforced weakly. The resulting nonlinear system of equations is solved monolithically with Newton’s method. The design sensitivities are computed by the adjoint method and the optimization problem is solved by a gradient-based algorithm. The characteristics of this optimization framework are studied with two-dimensional problems at steady state. Numerical results indicate that the proposed treatment of free-floating volumes introduces a discontinuity in the design evolution, yet the method is still successful in converging to meaningful designs.     

Adhesive surface design using topology optimization

Tailoring adhesive properties between surfaces is of great importance for micro-scale systems, ranging from managing stiction in MEMS devices to designing wall-scaling gecko-like robots. A methodology is introduced for designing adhesive interfaces between structures using topology optimization. Structures subjected to external loads that lead to delamination are studied for situations where displacements and deformations are small. Only the effects of adhesive forces acting normal to the surfaces are considered. An interface finite element is presented that couples a penalty contact formulation and a Lennard–Jones model of van der Waals adhesive forces. Two- and three dimensional design optimization problems are presented in which adhesive force distributions are designed such that load-displacement curves of delaminating structures match target responses. The design variables describe the adhesive energy per area of the interface between the surfaces, as well as the geometry of the delaminating structure. A built-in length scale in the formulation of the adhesion forces eliminates the need for filtering to achieve comparable optimal adhesive designs over a range of mesh densities. The resulting design problem is solved by gradient based optimization algorithms evaluating the design sensitivities by the adjoint method. Results show that the delamination response can be effectively manipulated by the method presented. Varying simultaneously both adhesive and geometric parameters yields a wider range of reachable target load-displacement curves than in the case varying adhesive energy alone.     

Improved genetic algorithm with two-level approximation using shape sensitivities for truss layout optimization

Truss layout optimization is a procedure for optimizing truss structures under the combined influence of size, shape and topology variables. This paper presents an Improved Genetic Algorithm with Two-Level Approximation (IGATA) that uses continuous shape variables and shape sensitivities to minimize the weight of trusses under static or dynamic constraints. A uniform optimization model including continuous size/shape variables and discrete topology variables is established. With the introduction of shape sensitivities, the first-level approximations of constraint functions are constructed with respect to shape/topology/size variables. This explicit problem is solved by implementation of a real-coded GA for continuous shape variables and binary-coded GA for 0/1 topology variables. Acceleration techniques are used to overcome the convergence difficulty of the mixed-coded GA. When calculating the fitness value of each member in the current generation, a second-level approximation method is embedded to optimize the continuous size variables effectively. The results of numerical examples show that the usage of continuous shape variables and shape sensitivities improves the algorithm performance significantly.     

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bézier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments. This research has been supported by the Austrian Science Fund FWF within the SFB “Numerical and Symbolic Scientific Computing” under the grant SFB F013, subprojects F1309 and F1315, by the Czech Ministry of Education under the grant AVČR 1ET400300415, by the Czech Grant Agency under the grant GAČR 201/05/P008 and by the Slovak Grant Agency under the project VEGA 1/0262/03.