Achieving stress-constrained topological design via length scale control

A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non‐linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user‐defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.

This paper focuses on topology optimization for additive manufacturing. In order to ensure that the optimized design is immediately manufacturable, it is essential to take into account the appropriate geometric constraints during the optimization. Two important constraints are minimum length scale and maximum overhang angle. A minimum length scale is needed to ensure that the condition on minimal printable feature sizes is satisfied, while an imposed overhang angle eliminates the need for a temporary support structure. This paper first shows that both constraints cannot simultaneously be met by a straightforward coupling of existing methods for length scale and overhang angle control. Next, a new filtering scheme is introduced, based on a specific combination of spatial filters, which allows direct control over these constraints in a minimum compliance topology optimization problem. A 2D benchmark problem and a complex 3D case study are presented to demonstrate that the proposed filtering scheme successfully imposes a target length scale in both the solid and the void phase of the design domain, while simultaneously allowing control over the overhang angle.

(Abstract) Techniques for imposing a minimum length scale, or minimum feature size, in continuum topology optimization have been proposed in literature, with the motivation being stabilization of the maximum stiffness problem and satisfying manufacturing constraints. Imposing an upper bound on feature sizes, however, has not been investigated. This paper proposes a scheme for restricting the maximum length scale in topology optimization. In short, the design domain is searched and structural features larger than the prescribed maximum length scale are penalized. The scheme is implemented in the context of minimum compliance design together with an existing minimum length scale methodology. The resulting optimization problem is continuous and solved using the Method of Moving Asymptotes (MMA). Beam design examples are considered and solutions are shown to be near 0/1 (void/solid) topologies that satisfy the minimum and maximum length scale criteria. The designer thus gains complete control over member sizes and therefore additional influence over cost and manufacturability. Further, restricting maximum length scale can potentially be used to introduce structural redundancy into the design, as loads previously carried by few large members are often redistributed over an interconnected system of smaller members.

This paper focusses on topology optimization for additive manufacturing. Two manufacturing constraints are considered: minimum length scale and maximum overhang angle. The first is needed to ensure that the condition on minimal printable feature sizes is satisfied, while the second eliminates the need for a temporary support structure. Filtering schemes have been proposed in the literature to ensure either a minimum length scale or a maximum overhang angle, but not both simultaneously. In this paper, it is shown that both constraints cannot simultaneously be met by simply applying both filters sequentially, as the effect of the first filter is destroyed by the second. A new, slightly more complex filtering scheme is therefore proposed, which does allow simultaneous control over length scale and overhang angle in a minimum compliance topology optimization problem. The method is successfully applied to a 2D benchmark problem.

Minimum length scale in topology optimization by geometric constraints

Piecewise length scale control for topology optimization with an irregular design domain

The optimization of the breech bus bar has an important influence on the lightweight research of the electromagnetic railgun. A topology optimization model considering volume constraints is established for the breech flat-plate bus bar. The bus bar is fed through sixteen coaxial cables. The optimal structure of the bus bar is calculated and given under the requirement of 50% volume constraint. In addition, the influence of four control parameters is also analyzed. These four parameters are respectively the prescribed volume fraction, the positioning dimension of cable arrangement, the filter radius, and the penalization power. For the first parameter, the optimal structures of the bus bar are given and compared when it is respectively 0.4, 0.6, 0.7, and 0.8. For the second parameter, the results show that the larger the positioning dimension, the smaller the size of all parts of the bus bar. For the third parameter, it is suggested to use the surrounding two-layer elements of each element to conduct sensitivity filtering. For the fourth parameter, it is reasonable to choose 3.0 as the penalization power value. It is meaningful for the structural design of the bus bar in the initial design stage.

This thesis presents a topology optimization methodology for the systematic design of optimal multifunctional silicon anode structures in lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such, this work considers two design objectives of minimum compliance under design dependent volume expansion, and maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the iteration history, mesh independence, and influence of prescribed volume fraction and minimum length scale are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the compliance and conduction design criteria. A weighting method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. Furthermore, a systematic parameter study is undertaken to determine the influence of the prescribed volume fraction and minimum length scale on the optimal combined topologies. The developments presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

Although compliant mechanism design is a thoroughly studied field, surprisingly little information can be found in literature regarding selection of optimal materials. This paper is intended to fill this gap. Density-based, geometrically robust, stress constrained topology optimization based on a total Lagrangian FEM formulation is used for investigation of a compliant inverter and a compliant gripper example. Changes in handling of the projection parameters and the stress constraints are proposed for improved algorithmic stability and accurate representation of the stresses and material stiffness in topology optimization. Large-scale optimization studies are carried out, varying elastic modulus and allowable stress in a wide range. A comparison with an Ashby chart shows the characteristics of the best suited material. It is shown, that an optimal Young’s modulus and a minimum required material strength (depending on the modulus) can be identified from the results. Altering critical optimization parameters, e.g. allowable volume fraction and the minimum length scale, their influence on the optimal material choice is investigated. Guidelines for compliant mechanism designers for efficient selection of suited materials are developed.

The objective of this investigation is to carry out the topology design optimization of the pickup suspension of high-speed optical pickups in DVD’s or CD ROM’s. The specific design target here is to increase the torsional eigenfrequency of the pickup as high as possible while keeping the focusing eigenfrequency between 25 Hz and 35 Hz, and the tracking eigenfrequency between 43 Hz and 57 Hz. These frequency ranges are set to meet other design constraints such as servo-controllability. A typical pickup consists of a bobbin having an optical lens and suspension plates to support it. Since the eigenfrequencies of the bobbin itself are in the order of 10 kHz, it can be modeled as a point mass when dealing with the low-frequency dynamic characteristics in consideration. These modeling techniques are verified in the previous researchers. In modeling the bobbin, we also consider its moments of inertia. In designing the suspension system, damping bond is usually applied to control the quality factors at the focusing and tracking frequencies. The location of the damping treatment on the suspension system may be determined by an optimization method, but this is not pursued in this work since the result by the earlier work can be applied to the present suspension system. Unlike static problems such as compliance minimization problems with single and multiple loads or compliant mechanism problems with single and multiple input/outputs, optimization problems involving eigenvalues suffer from mode-switching, which often causes serious difficulties in solution convergence. Mode-switching represents the phenomenon that eigenmodes are changed during the update of design variables. If the design target were simply to maximize a few lowest eigenfrequencies, the mode-switching problem would be handled rather easily. However, the present optimization problem is not the case because we must follow the eigenfrequencies of the specific modes such as tracking, focusing and twisting modes during the optimization process. To handle this situation satisfactorily, we should employ a technique to track the desired modes. In the present investigation, the desired modes are traced by means of the modal assurance criterion (MAC), which has been widely used in experimental modal analysis. For the pickup consisting of a bobbin and suspension plates, the present design goal can be achieved by optimizing the configuration of the suspension plates. To find an optimal plate configuration, we employ the topology optimization method, but the specific method employed here is the multiscale topology optimization method. In the multiscale optimization method, the density design variables are transformed into multiscaled design variables by a series of nonlinear and linear transformations. With the multiscale topology optimization method, the optimization can be completed more efficiently than the optimization by 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002, Atlanta, Georgia AIAA 2002-5483 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. the standard single-scale direct density method. As an optimizer, the feasible direction method is employed to handle effectively multiple constraints on eigenfrequencies. By the present application of the topology optimization method, the twisting eigenfrequency has been maximized while the focusing and tracking eigenfrequencies are kept in the prescribed frequency range. The optimized suspension plate in the present work outperforms the designs improved from the existing rectangular bend-type plates.

The topology optimization of continuum structures is investigated to obtain an optimized topology satisfying the fail-safe design principle with suitable redundancy components. An optimization model with a nonlinear objective and linear constraints is established by defining the reciprocal topology variable as design variables and linearly approximating the structural performance. The model is solved by a dual sequential quadratic programming (DSQP) algorithm. Topology optimization with displacement constraints is used as an example. The presented optimization model and solution approach offers the following advantages. (1) A volume constraint, which makes determining reasonable values difficult, need not be specified. (2) Weighted coefficients, which combine multiple compliances under different load cases into a combined compliance, need not be specified as well. (3) The presented optimization model is a type of single-objective programming and thus avoids the challenges in the min-max model. (4) The proposed method shows a strong capability of finding optimum solutions.

This article proposes an uncertainty‐oriented double‐scale topology optimization method considering macroreliability limitation and micromanufacturing control. The procedure combines the homogenization method to solve the equivalent elastic property of the microstructure, the feature distance theory to evaluate reliability, a density projection for manufacturability control, and the solid isotropic material with penalization model to suppress intermediate density. To coupling the micro–macro scale, the equivalent elastic property of the cell microstructure is evaluated by the numerical homogenization method and then endowed to the macroelement for finite element analysis. Uncertainty factors existed in optimization parameters are evaluated by the interval convex set model. By utilizing the interval parameter vertex method and the feature distance theory, the reliability of displacement constraint in the optimization model is evaluated and constrained. In terms of length scale control, the microdensity design variables are projected by a threshold function to obtain a clear microstructure that satisfies the preset minimum length scale constraint. Finally, three numerical examples are presented to illustrate the effect of micromanufacturing control, as well as the necessity of considering parameter uncertainty.

Traditional methods based on an element-wise parameterization of the material distribution applied to the topology optimization of fluidic systems often suffer from slow convergence of the optimization process, as well as robustness issues at increased Reynolds numbers. The local influence of the design variables in the traditional approaches is seen as a possible cause for the slow convergence. Non-smooth material distributions are suspected to trigger premature onset of instationary flows which cannot be treated by steady-state flow models. In the present work, we study whether the convergence and the versatility of topology optimization methods for fluidic systems can be improved by employing a parametric level-set description. In general, level-set methods allow controlling the smoothness of boundaries, yield a non-local influence of design variables, and decouple the material description from the flow field discretization. The parametric level-set method used in this study utilizes a material distribution approach to represent flow boundaries, resulting in a non-trivial mapping between design variables and local material properties. Using a hydrodynamic lattice Boltzmann method, we study the performance of our level-set approach in comparison to a traditional material distribution approach. By numerical examples, the parametric level-set approach is validated through comparison with traditional material distribution based methods. While the parametric level-set approach leads to similar optimal designs, the present study reveals no general improvements of the convergence of the optimization process and of the robustness of the nonlinear flow analyses when compared to the traditional material distribution approach. Instead, our numerical experiment suggests that a continuation method operating on the volume constraint is needed to achieve optimal designs at higher Reynolds numbers.

SummaryThe velocity field level‐set topological shape optimization method combines the implicit representation in the standard level‐set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level‐set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two‐ and three‐dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.

When applying the sequential quadratic programming (SQP) algorithm to topology optimization, using the quasi-Newton methods or calculating the Hessian matrix directly will result in a considerable amount of calculation, making it computationally infeasible when the number of optimization variables is large. To solve the above problems, this paper creatively proposes a method for calculating the approximate Hessian matrix for structural topology optimization with minimum compliance problems. Then, the second-order Taylor expansion transforms the original problem into a series of separable and easy-to-solve convex quadratic programming (QP) subproblems. Finally, the quadratic programming optimality criteria (QPOC) method and the QP solver of MATLAB are used to solve the subproblems. Compared with other sequential quadratic programming methods, the advantage of the proposed method is that the Hessian matrix is diagonally positive definite and its calculation is simple. Numerical experiments on an MBB beam and cantilever beam verify the feasibility and efficiency of the proposed method.