Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization

We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.     

A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

Constraint force design method for topology optimization of planar rigid-body mechanisms

This study develops a new design method called the constraint force design method, which allows topology optimization for planar rigid-body mechanisms. In conventional mechanism synthesis methods, the kinematics of a mechanism are analytically derived and the positions and types of joints of a fixed configuration (hereafter the topology) are optimized to obtain an optimal rigid-body mechanism tracking the intended output trajectory. Therefore, in conventional methods, modification of the configuration or topology of joints and links is normally considered impossible. In order to circumvent the fixed topology limitation in optimally designing rigid-body mechanisms, we present the constraint force design method. This method distributes unit masses simulating revolute or prismatic joints depending on the number of assigned degrees of freedom, analyzes the kinetics of unit masses coupled with constraint forces, and designs the existence of these constraint forces to minimize the root-mean-square error of the output paths of synthesized linkages and a target linkage using a genetic algorithm. The applicability and limitations of the newly developed method are discussed in the context of its application to several rigid-body synthesis problems.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

Application of topology optimization to design an electric bicycle main frame

Electric bicycle main frame is the most principal structure, connecting and supporting other various components, while bearing a variety of forces and moments. In this paper the topology optimization technology is applied to generate robust electric bicycle main frame by optimizing the material distribution subject to the constraints and dynamic loads. Geometric, mechanical and finite element models, as well as a flexible coupling dynamic model are constructed. Validity and accuracy of these models are investigated through real-life testing. By applying typical road excitation, dynamic loads of all key points are extracted. A set of forces data is extracted every 0.5?s during the whole simulation, including peak values of these forces. In order to obtain appropriate topology optimization results, the values of two crucial parameters, volume fraction and minimum member size, are discussed respectively. Then the topology optimization of multi-load case is implemented with the objective of minimizing the set of weighted compliances resulting from individual load cases. Results illustrate that element density distribution of the model is optimized with manufacturing constraints of minimum member size control and extrusion constraint. Consequently, the better frame form design of the electric bicycle is obtained. Modal analysis for the original and refined models is performed respectively to evaluate the structure stiffness. The results indicate that this optimization program is effective enough to develop a new electric bicycle frame as a reference for manufacturers.     

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.     

A new overhang constraint for topology optimization of self-supporting structures in additive manufacturing

This work falls within the scope of computer-aided optimal design, and aims to integrate the topology optimization procedures and recent additive manufacturing technologies (AM). The elimination of scaffold supports at the topology optimization stage has been recognized and pursued by many authors recently. The present paper focuses on implementing a novel and specific overhang constraint that is introduced inside the topology optimization problem formulation along with the regular volume constraint. The proposed procedure joins the design and manufacturing processes into a integrated workflow where any component can directly be manufactured with no requirement of any sacrificial support material right after the topology optimization process. The overhang constraint presented in this work is defined by the maximum allowable inclination angle, where the inclination of any member is computed by the Smallest Univalue Segment Assimilating Nucleus (SUSAN), an edge detection algorithm developed in the field of image analysis and processing. Numerical results on some benchmark examples, along with the numerical performances of the proposed method, are introduced to demonstrate the capacities of the presented approach.     

Cost optimization of industrial steel building structures

The paper presents the simultaneous cost, topology and standard cross-section optimization of single-storey industrial steel building structures. The considered structures are consisted from main portal frames, which are mutually connected with purlins. The optimization is performed by the mixed-integer non-linear programming approach, MINLP. The MINLP superstructure of different structure/topology and standard cross-section alternatives has been generated and the MINLP optimization model of the structure has been developed. The defined cost objective function is subjected to the set of (in)equality constraints known from the structural analysis. Internal forces and deflections are calculated by the elastic first-order analysis constraints. The dimensioning constraints of steel members are defined in accordance with Eurocode 3. The modified outer-approximation/equality-relaxation (OA/ER) algorithm, a two-phase MINLP strategy and a special prescreening procedure of discrete alternatives are used for the optimization. A numerical example of the cost optimization of a single-storey industrial steel building is presented at the end of the paper.     

Topological derivative-based topology optimization of structures subject to design-dependent hydrostatic pressure loading

In this paper the topological derivative concept is applied in the context of compliance topology optimization of structures subject to design-dependent hydrostatic pressure loading under volume constraint. The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations, such as holes, inclusions, source-terms and cracks. In particular, the topological asymptotic expansion of the total potential energy associated with plane stress or plane strain linear elasticity, taking into account the nucleation of a circular inclusion with non-homogeneous transmission condition on its boundary, is rigorously developed. Physically, there is a hydrostatic pressure acting on the interface of the topological perturbation, allowing to naturally deal with loading-dependent structural topology optimization. The obtained result is used in a topology optimization algorithm based on the associated topological derivative together with a level-set domain representation method. Finally, some numerical examples are presented, showing the influence of the hydrostatic pressure on the topology of the structure.     

Topology optimization of the primary mirror of a multi-spectral camera

A study of the topology optimization of a multi-spectral camera for space-use is presented. The optimization is carried out under self-weight and polishing pressure loading. As an objective function, a measure of Strehl ratio is used. Total mass of the primary mirror is given as a constraint to the optimization problem. The sensitivities of the objective function and constraint are calculated by direct differentiation method. The optimization procedure is carried out by an optimality criteria method. For the light-weight primary mirror design, a three dimensional model is treated. As a preliminary example, topology optimization considering a self-weight loading is treated. In the second example, the polishing pressure is also included as a loading in the topology optimization of the mirror. Results of the optimized design topology for the mirror with various mass constraints are presented. Received September 3, 2001 RID="*" ID="*"A shorter version of this paper was presented at ISSMO’s Fourth World Congress in 2001, in Dalian, China. Dr Park was awarded for it the ISSMO/Springer prize for young scientists-2001