An evaluative study on ESO and SIMP for optimising a cantilever tie—beam

We examine both the evolutionary structural optimisation (ESO) and solid isotropic microstructure with penalisation (SIMP) methodologies by investigating a cantilever tie–beam. Initially, both ESO and SIMP produce designs with higher objective function values relative to a previously published ‘intuitive’ design. However, after careful investigation of the numerical parameters such as the initial design domain and the mesh size, both methods obtain designs that have lower objective function values relative to the intuitive design. Thus, a clearer understanding of the numerical parame- ters and their influence on optimisation methods is achieved.     

The SRV constraint for 0/1 topological design

In density-based topological design, 0/1 solutions are often sought, that is, one expects that the final design includes either elements with full material or no material, excluding grey areas. The accepted technique for achieving binary values for the densities is to use a solid isotropic microstructure with penalization (SIMP) material for which Young’s modulus is a polynomial function of the otherwise continuous relative densities. This approach indeed enhances 0/1 solutions in a significant manner and as such it has achieved prominent status in topological design. Nevertheless, this paper proposes a possible alternative to the SIMP methodology for generating 0/1 structures. The design variables are still the densities of the finite elements but Young’s modulus is a linear function of these densities (in some sense, a SIMP material without penalty). In order to drive the solution to a 0/1 layout a new constraint, labeled the sum of the reciprocal variables (SRV), is introduced. The constraint stipulates that the SRV must be larger or equal to its value at a discrete design for a specified amount of material. It is understood that this implies a minimum gage on the design variables, a provision which is also present in the standard fixed-grid formulation to avoid singular stiffness matrices. The technique turned out to be very effective in conjunction with the method of moving asymptotes (MMA) when using topological design methods for finding optimal layouts of patches of piezo-electric (PZT) material in order to minimize the mechanical noise emanating from vibrating surfaces. It also performed satisfactorily in classical structural topological design instances, as can be seen in the numerical examples that illustrate this work.     

A hybrid topology optimization methodology combining simulated annealing and SIMP

Structural topologies obtained by SIMP are not completely satisfactory since they exhibit areas with intermediate densities which lack physical meaning. Thus, the designer is left with the nontrivial decision of rounding and approximating the solution. This paper presents SA–SIMP, a hybrid methodology that couples simulated annealing and SIMP under a look-ahead scheme that gradually fixes or removes elements with intermediate densities (i.e., gray areas in SIMP). After designing several strategies and testing them on different structures, SA–SIMP consistently led to topologies with both lower compliances and less gray areas than those obtained by SIMP alone.     

Stress-related topology optimization via level set approach

Considering stress-related objective or constraint functions in structural topology optimization problems is very important from both theoretical and application perspectives. It has been known, however, that stress-related topology optimization problem is challenging since several difficulties must be overcome in order to solve it effectively. Traditionally, SIMP (Solid Isotropic Material with Penalization) method was often employed to tackle it. Although some remarkable achievements have been made with this computational framework, there are still some issues requiring further explorations. In the present work, stress-related topology optimization problems are investigated via a level set-based approach, which is a different topology optimization framework from SIMP. Numerical examples show that under appropriate problem formulations, level set approach is a promising tool for stress-related topology optimization problems.     

An isogeometrical approach to structural topology optimization by optimality criteria

The Isogeometric Analysis (IA) method is applied for structural topology optimization instead of finite elements. For this purpose, a control point based Solid Isotropic Material with Penalization (SIMP) method is employed and the material density is considered as a continuous function throughout the design domain and approximated by the Non-Uniform Rational B-Spline (NURBS) basis functions. To prevent the formation of layouts with porous media, a penalization technique similar to the SIMP method is used. For optimization an optimality criteria is derived and implemented. A few examples are presented to demonstrate the performance of the method. It is shown that, dissimilar to the element based SIMP topology optimization, the resulted layouts by this method are independent of the number of the discretizing control points and checkerboard free.     

Topology optimization for light-trapping structure in solar cells

The limitation associated with the low optical absorption remains to be the main technical barrier that constrains the efficiency of thin–film solar cells in energy conversion. Effective design of light-trapping structure is critical to increase light absorption, which is a highly complex phenomenon governed by several competing physical processes, imposing a number of challenges to topology optimization. This paper presents a general, yet systematic approach exploiting topology optimization for designing highly efficient light-trapping structures. We first demonstrate the proposed approach using genetic algorithm (GA) based non-gradient topology optimization (NGTO), which is robust for achieving highly-efficient designs of slot-waveguide based cells with both low-permittivity and high-permittivity scattering material at single wavelength or over a broad spectrum. The optimized light-trapping structure achieves a broadband absorption efficiency of 48.1 % and more than 3-fold increase over the Yablonovitch limit. The fabrication feasibility of the optimized design is also demonstrated. Next, the gradient topology optimization (GTO) approach for designing light-trapping structure is explored based on the Solid Isotropic Material with Penalization (SIMP) method. Similar designs are obtained through both GA based NGTO and SIMP based GTO, which verifies the validity of both approaches. Insights into the application of both approaches for solving the nanophotonic design problem with optimization nonlinearity are provided.     

Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.     

Multi-material topology optimization using ordered SIMP interpolation

In this paper an ordered multi-material SIMP (solid isotropic material with penalization) interpolation is proposed to solve multi-material topology optimization problems without introducing any new variables. Power functions with scaling and translation coefficients are introduced to interpolate the elastic modulus and the cost properties for multiple materials with respect to the normalized density variables. Besides a mass constraint, a cost constraint is also considered in compliance minimization problems. A heuristic updating scheme of the design variables is derived from the Kuhn-Tucker optimality condition (OC). Since the proposed method does not rely on additional variables to represent material selection, the computational cost is independent of the number of materials considered. The iteration scheme is designed to jump across the discontinuous point of interpolation derivatives to make stable transition from one material phase to another. Numerical examples are included to demonstrate the proposed method. Because of its conceptual simplicity, the proposed ordered multi-material SIMP interpolation can be easily embedded into any existing single material SIMP topology optimization codes.     

柔性变形机翼后缘拓扑优化设计

为了实现机翼表面的自适应变形和结构轻量化,将柔件机构引入到机翼后缘形状变化结构设计中.应用连续体拓扑优化技术,以实际位移与目标位移之间的偏差为目标函数,材料用量为约束,建立SIMP(solid isotropic material with penalization)密度刚度插值模型.采用Matlab编程对柔性机构进行了优化设计,并对不同参数下的优化结果进行了讨论,最后进行机构的仿真分析.研究结果显示该柔性机构能够实现预期的形状变化,证明了方法的正确性,为柔性机翼设计提供理论基础.     

A topology optimization approach using VOF method

Topology optimization methods with continuous design variables obtained by the homogenization formula or the solid isotropic microstructure with penalty (SIMP) model are widely used in the layout of structures. In the implementation of these approaches, one must take into account several issues, e.g., irregularity of the problem, occurrence of the checkerboard pattern, and intermediate density. To suppress these phenomena, the employment of additional strategies such as the perimeter control or the filtering method will be required. In this paper, a topology optimization method which can eliminate these difficulties is developed based on the volume of fluid (VOF) method. In the method, shape design is described in terms of the VOF function. Since this function is defined by a volume fraction of material occupying each element, it can be recognized as a continuous material density in the SIMP model. Within the framework of the VOF analysis, the topology optimization procedure is reduced to a convection motion of the material density governed by a Hamilton–Jacobi equation as in the level set method. Through numerical examples, the validity of the proposed method is investigated.     

Combined topology and fiber path design of composite layers using cellular automata

The popular Solid Isotropic Material Penalization (SIMP) technique of topology design is extended to simultaneous fiber-angle and topology design of composite laminae in a cellular automata (CA) framework. CA is a novel methodology to simulate a physical phenomenon based on iterative local updates of both field and design variables. Displacements are updated satisfying local equilibrium of CA cells. Fiber angles and density measures are updated based on the optimality criteria for the minimum compliance design. Numerical results for the design of 2D cantilever plates for single and multiple load cases are used to demonstrate the robustness of the proposed algorithm.     

Topology optimization for hybrid additive-subtractive manufacturing

Hybrid additive-subtractive manufacturing is gaining popularity by making full use of geometry complexity produced by additive manufacturing and dimensional accuracy derived from subtractive machining. Part design for this hybrid manufacturing approach has been done by trial-and-error, and no dedicated design methodology exists for this manufacturing approach. To address this issue, this work presents a topology optimization method for hybrid additive and subtractive manufacturing. To be specific, the boundary segments of the input design domain are categorized into two types: (i) Freeform boundary segments freely evolve through the casting SIMP method, and (ii) shape preserved boundary segments suppress the freeform evolvement and are composed of machining features through a feature fitting algorithm. Given the manufacturing strategy, the topology design is produced through additive manufacturing and the shape preserved boundary segments will be processed by post-machining. This novel topology optimization algorithm is developed under a unified SIMP and level set framework. The effectiveness of the algorithm is proved through a few numerical case studies.     

On the influence of model reduction techniques in topology optimization of flexible multibody systems using the floating frame of reference approach

Employing the floating frame of reference formulation in the topology optimization of dynamically loaded components of flexible multibody systems seems to be a natural choice. In this formulation the deformation of flexible bodies is approximated by global shape functions, which are commonly obtained from finite element models using model reduction techniques. For topology optimization these finite element models can be parameterized using the solid isotropic material with penalization (SIMP) approach. However, little is known about the interplay of model reduction and SIMP parameterization. Also securing the model reduction quality despite major changes of the design during the optimization has not been addressed yet. Thus, using the examples of a flexible frame and a slider-crank mechanism this work discusses the proper choice of the model reduction technique in the topology optimization of flexible multibody systems.     

Design methodology of piezoelectric energy-harvesting skin using topology optimization

This paper describes a design methodology for piezoelectric energy harvesters that thinly encapsulate the mechanical devices and exploit resonances from higher-order vibrational modes. The direction of polarization determines the sign of the piezoelectric tensor to avoid cancellations of electric fields from opposite polarizations in the same circuit. The resultant modified equations of state are solved by finite element method (FEM). Combining this method with the solid isotropic material with penalization (SIMP) method for piezoelectric material, we have developed an optimization methodology that optimizes the piezoelectric material layout and polarization direction. Updating the density function of the SIMP method is performed based on sensitivity analysis, the sequential linear programming on the early stage of the optimization, and the phase field method on the latter stage of the optimization to obtain clear optimal shapes without intermediate density. Numerical examples are provided that illustrate the validity and utility of the proposed method.     

Design of absorbing material distribution for sound barrier using topology optimization

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0–1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton–Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.     

Regularization of shape optimization problems using FE-based parametrization

This paper introduces a general fully stabilized mesh based shape optimization strategy, which allows for shape optimization of mechanical problems on FE-based parametrization. The well-known mesh dependent results are avoided by application of filter methods and mesh regularization strategies. Filter methods are successfully applied to SIMP (Solid Isotropic Material with Penalization) based topology optimization for many years. The filter method presented here uses a specific formulation that is based on convolution integrals. It is shown that the filter methods ensure mesh independency of the optimal designs. Furthermore they provide an easy and robust tool to explore the whole design space with respect to optimal designs with similar mechanical properties. A successful application of optimization strategies with FE-based parametrization requires the combination of filter methods with mesh regularization strategies. The latter ones ensure reliable results of the finite element solutions that are crucial for the sensitivity analysis. This presentation introduces a new mesh regularization strategy that is based on the Updated Reference Strategy (URS). It is shown that the methods formulated on this mechanical basis result in fast and robust mesh regularization methods. The resulting grids show a minimum mesh distortion even for large movements of the mesh boundary. The performance of the proposed regularization methods is demonstrated by several illustrative examples.     

Topology optimization of free vibrating continuum structures based on the element free Galerkin method

In this paper, the topology optimization design of the free vibrating continuum structures is formulated based on the element free Galerkin (EFG) method. Considering the relative density of nodes as design variable, and the maximization of the fundamental eigenvalue as an objective function, the mathematical formulation of the topology optimization model is developed using the solid isotropic microstructures with penalization (SIMP) interpolation scheme. The topology optimization problem is solved by the optimality criteria method. Finally, the feasibility and efficiency of the proposed method are illustrated with several 2D examples that are widely used in the topology optimization design.     

Material interpolation schemes for unified topology and multi-material optimization

This paper presents two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation schemes originally developed for isotropic mixtures of two isotropic material phases. The new interpolation schemes provide generally applicable interpolation schemes between an arbitrary number of pre-defined materials with given (anisotropic) properties. The method relies on a large number of sparse linear constraints to enforce the selection of at most one material in each design subdomain. Topology and multi-material optimization is formulated within a unified parametrization.