A consistent approximation of the total perimeter functional for topology optimization algorithms

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

This paper deals with topology optimization seen from an industrial and a manufacturing point of view. What the industrial users see as key issues when implementing topology optimization are frequently not the topics that the academic environments consider when implementing topology optimization. One of the key issues is that topology optimization should be integrated into an existing CAD and CAE environment (example of FE-solvers: ABAQUS, ANSYS, CATOPO, NX Nastran, MSc Nastran, MSc Marc and Permas). Here, it is of importance to note that the models analyzed in the industrial CAE environments often consist of many finite elements (can be more than 2 million elements) and different types of elements (continuum, beam, membrane, shell, bar and rigid elements) which all can be employed in the same model. Furthermore, symmetry and manufacturing constraints are important for ensuring that the results of the topology optimized are manufacturable. Several examples will be given of how structural topology optimization can be used as an add-on module in existing CAE environments However, from an academic point of view an add-on module including topology optimization for commercial FE-solvers is also of interest because the latest finite element and solver technology can be applied in the modeling for the topology optimization. Furthermore, the aim is also to show where the industry is today and which demands exist for the future. The results reported here are based upon the work done by FE-Design and their industrial partners using the topology optimization module (TOSCA.topo) and smooth module (TOSCA.smooth) of the program TOSCA.

This paper discusses a geometrical optimization method for thermoelectric generators. Our method provides the optimized geometry in accord with various arbitrary conditions such as types of materials, volume of materials, temperatures of installation position, and shape of installation position. It has the potential to contribute to improving thermoelectric devices in supporting design stages. By establishing the coupled equations of state for the thermoelectric problem, an analytical model subject to these equations is introduced that mimics the closed circuit composed of the thermoelectric materials, electrodes and a resistor. The total electric power lost through the resistor is formulated as an objective function for optimization. The proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization (SIMP) method used in topology optimizations. Simple relations are formulated between the density function of SIMP method and physical properties of thermoelectric material. Sensitivity analysis for the objective function is formulated with respect to the density function and the adjoint equations required for calculating it. Depending on the sensitivity, the density function is updated using the method of moving asymptotes, which has numerous benefits in various optimization problems by virtue of its combination with topology optimization. Finally, the numerical examples are provided to demonstrate the validity of our method.

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

Most existing study on topology optimization have concentrated on maximizing the system stiffness. Especially, the minimization of static compliance subject to the volume fraction is widespread in the formulation. From the engineering point of view, structural strength design is of vital importance. Past study on stress constraint have shown that an amount of numerical difficulties with the stress-constrain topology optimization exist including the so-called singularity, vast of stress constraints, highly nonlinear behavior and so on. To achieve the topological design under stress constraint requirement, the normalized stress measure using p-norm function is adopted for the reduction of stress constraints. Following the modeling manner of independent continuous mapping method, the reciprocal function of relative density is regarded as the design variables. The sensitivities of stress constraint and volume objective with respect to the design variable are derived, and their explicit expressions are formulated based on the first-order and second-order Taylor approximation respectively. By setting up the sub-problem in the form of a quadratic program, the original topology optimization problem is efficiently solved using the sequential quadratic programming approach. The difference between stiffness and strength design, as well as the effect of various upper bounds of stress value on the optimized results for stress constraint are investigated in 2D numerical examples. Through the comparison of the proposed method and traditional variable density method, the feasibility and effectiveness of the proposed optimization approach in stress constrained problems are verified. The results also demonstrate that the consideration of stress constraint in continuum structure is indispensable.

Explicit and efficient topology optimization for three-dimensional structures considering geometrical nonlinearity

In this thesis numerical methods combining evolutionary optimization algorithms and finite element analysis are developed and applied to problems in cardiac mechanics and structural optimization. In the first application finite element analysis is combined with a real encoded genetic algorithm to determine unknown material parameters in constitutive model for ventricular myocardium using an inverse approach. As a second application suppression of global instabilities in minimum-weight Michell structures is investigated. A real encoded genetic algorithm is combined with a finite element eigenvalue buckling analysis for suppressing global buckling associated with minimum-weight structures. A novel finite element based evolutionary algorithm is developed to determine optimal structural topologies for a 2D continuum. Finally, the topology optimization algorithm described above is modified to determine optimal structural topology for design domains composed of materials having different strengths in tension and compression (dual strength materials). Optimal topologies evolve through successive removal and admission of material based on the current strain energy distribution. Illustrative examples demonstrate the ability to determine optimal structural configuration for a dual strength material domain. Optimal topologies obtained using current scheme are shown to agree with minimum-volume layouts generated using analytic methods based on dual strength material optimality criteria.

A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

Soft molecular electrostatic potentials (SMEP or SEMP) have been recently defined substituting the point-like proton by a Gaussian positive charge distribution. In the present paper an additional step is taken forward, transforming SMEP into a completely soft MEP (CoSMEP). Such transformation is carried out using a charge distributed proton as in SMEP and also a Gaussian positive nuclear charge distribution, instead of the classical point-like nuclear charges. The general form of MEP is roughly preserved, but new features can be noticed. Such new point of view is also associated to the possibility to redefine the molecular charge density. Definition of CoSMEP is thus connected to the definition of total molecular density functions (DF), where to the negative electronic DF is summed up the soft nuclear DF, made of linear combinations of Gaussian distributions of nuclear charges.

A fully developed flow thermofluid model for topology optimization of 3D-printed air-cooled heat exchangers

Topology optimization is the procedure of determining the connectivity, shape and location of voids inside a given design domain. This chapter proposes a quantum-inspired evolutionary algorithm (QEA) for structural topology optimization. It reviews the formulation of the continuum structural topology optimization model and the characteristics of quantum computing. The chapter also proposes the detailed implementation of a QEA-based topology optimization. It discusses two issues: the density-based continuum structural topology optimization formulation and the characteristics of quantum computing, which may be conducive to reducing the computational expense of topology optimization. The entire theoretical procedure of quantum-inspired topology optimization, which mainly combines FEM analysis, design sensitivity solution and quantum-inspired search, is accomplished for continuum structures. As the experimental implementation of quantum computation needs initialization, coherent manipulation, control and read of the fragile quantum system, practically building quantum computers has proved extremely difficult.

The production of complex functional structures becomes possible using Additive Manufacturing (AM) technology. Topology optimization allows to obtain lightweight and high-performance parts. In this study, the combining of topology optimization with additive manufacturing is proposed. This combination is mainly motivated by two principal reasons. Firstly, topology optimization generates complex structures that cannot be achieved by conventional processes. Secondly, a great flexibility in part design is provided by additive manufacturing. Indeed, an automated algorithm for the topology optimization of structures obtained by additive manufacturing is presented. The proposed methodology is based on the generation of the geometry and the mesh of the studied structure using ANSYS software that uses finite element analysis (FEA). Then, the topology optimization algorithm is implemented in MATLAB. The developed algorithm is illustrated by a numerical example involving the minimization of the compliance of 3D beam structure.

An integrated method of topological optimization and path design for 3D concrete printing

In this paper, density based topology optimization is considered in nonlinear magnetostatics. Previously, topology optimization in linear magnetostatics was formulated to design Hall-effect thrusters and it was solved using gradient based solvers. In that previous version, the gradient information was provided by the continuous adjoint method. Herein, the adjoint approach is updated in order to take into account the saturation aspect of materials. This new version of the adjoint method is implemented and compared to a finite difference approach. The comparison shows that the so-developed adjoint based gradient information in non linear magnetostatics is accurate and ready for use to solve a topology optimization design problem. Hence, a Hall-effect thruster topology optimization problem is solved in both linear and non-linear cases and a comparative description of the found designs will be presented.

In this paper, a method is proposed for the design optimization of structural components where both its shape and topology are optimized. The boundaries of the shape of the structure are represented using contours of a “shape density” function and the shape is optimized using a sequential approximate optimization algorithm. The contour of the density function corresponding to a threshold value is defined as the boundary of the shape. The shape density function is defined over a feasible domain and is represented by a continuous piece-wise interpolation over the finite elements used for structural analysis. The values of the density function at the nodes serve as the design variables of the optimization problem. The advantage of this shape representation is that both shape and topology of the structure can be modified and optimized by the optimization algorithm. The material properties of the structure are assumed to depend on the density function. Many approximate material property-density relations have been studied. The shape and topology of structural components are optimized with the objective of minimizing the compliance subject to a constraint on the total mass of the structure. With the advent of novel layered manufacturing techniques, manufaturing no longer limits the feasibility of some of the complex shapes and topologies thus designed.