Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.     

A coordinate descent based method for geometry optimization of trusses

The main objective of the structure optimization is to reduce the manufacturing cost of the structure. However, economical design should not deviate from engineering restrictions and laws. Based on this concept, various methods have been invented to optimize structures. In this research, some variations in one of the very simple and primary methods called “coordinate descent” or “successive coordinate search” is used for geometry optimization of the trusses. In order to increase the optimization speed and decrease the solution time of the problem, fast matrix analysis methods has been employed leading to reduction of the time required for calculating of the objective function. Furthermore, an optimized solution seeking algorithm has been introduced. In this algorithm, some factors have been defined by which the relation between the optimization time and these factors have been calculated, both experimentally and mathematically and the appropriate values of these factors are obtained to reduce the optimization time. Moreover, the effect of increasing the accuracy of the optimum solution on the number of analyses has been studied. Analysis of the presented method shows that regardless of its simplicity, it can be utilized as a fast method for optimizing variety of structures.     

Topology optimization for minimum compliance under multiple loads based on continuous distribution of members

A method to minimize the compliance of structures subject to multiple load cases is presented. Firstly, the material distribution in design domain is optimized to form a truss-like continuum. The anisotropic composite is employed as the material model to simulate the constitutive relation of the truss-like continuum. The member densities and orientations at the nodes are taken as design variables. The member densities and orientations at any point in an element vary continuously. Then, parts of members, which are formed according to the member distribution field, are chosen to form the nearly optimum discrete structure. Lastly, the positions of the nodes and the cross-sectional areas of the members are optimized. In the above process, numerical instabilities such as checkerboard and mesh dependencies disappear without any additional technique. The sensitivities of the compliance are derived. Examples are presented to demonstrate the capability of the proposed method.     

Topology and parameter optimization of a foaming jig reinforcement structure by the response surface method

A dilemma in the foaming of inner polyurethane (PU) pieces for household refrigerators is that of keeping the production costs down without adversely affecting the dimension precision. One way to do this is to reduce the electric power consumption spent running the polyurethane foaming line in which a number of heavy foaming jigs are continuously circulating, by optimally designing the reinforcement structure of the jig. In this paper, topology optimization and parameter optimization utilizing the response surface method (RSM) are applied for the optimum design of the jig reinforcement structure, in order to minimize the total jig weight while securing the dimension precision of the foamed urethane case. Both the reliability of the approximated response surfaces and the validity of the proposed optimization procedure are verified through illustrative optimization experiments. In addition, it is confirmed that the proposed procedure provides us with an optimum reinforcement structure which remarkably reduces the total jig weight.     

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.     

Topology optimization of heat conduction problems using the finite volume method

This note addresses the use of the finite volume method (FVM) for topology optimization of a heat conduction problem. Issues pertaining to the proper choice of cost functions, sensitivity analysis, and example test problems are used to illustrate the effect of applying the FVM as an analysis tool for design optimization. This involves an application of the FVM to problems with nonhomogeneous material distributions, and the arithmetic and harmonic averages have here been used to provide a unique value for the conductivity at element boundaries. It is observed that when using the harmonic average, checkerboards do not form during the topology optimization process. Preliminary results of the work reported here were presented at the WCSMO 6 in Rio de Janeiro 2005, see Gersborg-Hansen et al. (2005b).     

Topology optimization of frame structures—joint penalty and material selection

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.     

Topology optimization of shell structures using adaptive inner-front (AIF) level set method

A new topology optimization using adaptive inner-front level set method is presented. In the conventional level set-based topology optimization, the optimum topology strongly depends on the initial level set due to the incapability of inner-front creation during the optimization process. In the present work, in this regard, an algorithm for inner-front creation is proposed in which the sizes, the positions, and the number of new inner-fronts during the optimization process can be globally and consistently identified. In the algorithm, the criterion of inner-front creation for compliance minimization problems of linear elastic structures is chosen as the strain energy density along with volumetric constraint. To facilitate the inner-front creation process, the inner-front creation map is constructed and used to define new level set function. In the implementation of inner-front creation algorithm, to suppress the numerical oscillation of solutions due to the sharp edges in the level set function, domain regularization is carried out by solving the edge smoothing partial differential equation (smoothing PDE). To update the level set function during the optimization process, the least-squares finite element method (LSFEM) is adopted. Through the LSFEM, a symmetric positive definite system matrix is constructed, and non-diffused and non-oscillatory solution for the hyperbolic PDE such as level set equation can be obtained. As applications, three-dimensional topology optimization of shell structures is treated. From the numerical examples, it is shown that the present method brings in much needed flexibility in topologies during the level set-based topology optimization process.     

Design of an energy-absorbing structure using topology optimization with a multimaterial model

To accommodate the dual objectives of many engineering applications, one objective to minimize the mean compliance for the stiffest structure under normal service conditions and the other objective to maximize the strain energy for energy absorption during excessive loadings, topology optimization with a multimaterial model is applied to the design of an energy-absorbing structure in this paper. The effective properties of the three-phase material are derived using a spherical microinclusion model. The dual objectives are combined in a ratio formation. Numerical examples from the proposed method are presented and discussed.     

Optimal riser design in sand casting process by topology optimization with SIMP method I: Poisson approximation of nonlinear heat transfer equation

The optimal design of a casting feeding system is considered. The problem is formulated as the volume constrained topology optimization and is solved with the finite element analysis, explicit design sensitivity, and numerical optimization. In contrast to the traditional topology optimization where the objective function is defined on the design space, in the presented method, the design space is a subset of the complement of the objective function space. To accelerate optimization procedure, the nonlinear unsteady heat transfer equation is approximated with a Poisson-like equation. The feasibility of the presented method is supported with illustrative examples.