Applications of one-shot methods in PDEs constrained shape optimization

The paper deals with applications of numerical methods for optimal shape design of composite materials structures and devices. We consider two different physical models described by specific partial differential equations (PDEs) for real-life problems. The first application relates microstructural biomorphic ceramic materials for which the homogenization approach is invoked to formulate the macroscopic problem. The obtained homogenized equation in the macroscale domain is involved as an equality constraint in the optimization task. The second application is connected to active microfluidic biochips based on piezoelectrically actuated surface acoustic waves (SAWs). Our purpose is to find the best material-and-shape combination in order to achieve the optimal performance of the materials structures and, respectively, an improved design of the novel nanotechnological devices. In general, the PDEs constrained optimization routine gives rise to a large-scale nonlinear programming problem. For the numerical solution of this problem we use one-shot methods with proper optimization algorithms and inexact Newton solvers. Computational results for both applications are presented and discussed.     

Design of piezocomposite materials and piezoelectric transducers using topology optimization—Part I

Summary Currently developments of piezocomposite materials and piczoelectric actuators have been based on the use of simple analytical models, test of prototypes, and analysis using the finite element method (FEM), usually limiting the problem to a parametric optimization. By changing the topology of these devices or their components, we may obtain an improvement in their performance characteristics. Based on this idea, this paper discusses the application of topology optimization combined with the homogenization method and FEM for designing piezocomposite materials. The homogenization method allows us to calculate the effective properties of a composite material knowing its unit cell topology. New effective properties that improves the electromechanical efficiency of the piezocomposite material are obtained by designing the piezocomposite unit cell. This method consists of finding the distribution of the material and void phases in a periodic unit cell that optimizes the performance characteristics of the piezocomposite. The optimized solution is obtained using Sequential Linear Programming (SLP). A general homogenization method applied to piczoelectricity was implemented using the finite element method (FEM). This homogenization method has no limitations regarding volume fraction or shape of the composite constituents. The main assumptions are that the unit cell is periodic and that the scale of the composite part is much larger than the microstructure dimensions. Prototypes of the optimized piezocomposites were manufactured and experimental results confirmed the large improvement. Department of Mechanical Engineering and Applied Mechanics Department of Mechanical Engineering and Applied Mechanics     

Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.     

Simultaneous shape,topology, and homogenized properties optimization

In this brief note, we present an approach that combines the three classical techniques in structural optimization, i.e. the boundary variation and the topological and the homogenization methods. As a first test of this method, we apply it to the compliance opti-mization in .     

Topology optimization of hyperbolic metamaterials for an optical hyperlens

This paper presents a level set-based topology optimization method for the microstructural design of an optical hyperlens. The resolution of conventional optics is generally diffraction limited, but the diffraction limit can be overcome by a hyperlens that converts evanescent waves to propagation waves, utilizing a cylindrical geometry to magnify the subwavelength features of imaged objects, which allows such features to be resolved beyond the diffraction limit at the hyperlens output. To support electromagnetic waves that contain information of subwavelength features, the hyperlens must have material properties that include a positive permittivity in the angular direction and a negative permittivity in the radial direction. Here, an energy-based homogenization method is used to obtain the effective permittivity of a metamaterial hyperlens unit cell. A level set-based topology optimization is applied so that the boundaries of the structure are clearly expressed. Moreover, a finite element mesh regeneration scheme is used to precisely capture the boundaries of the metal and dielectric materials of the unit cell that will ultimately form the hyperlens. The optimization algorithm uses the finite element method (FEM) to solve the equilibrium and adjoint equations. Optimum design examples for the design of a hyperlens microstructure are provided to confirm the utility and validity of the presented method.     

Optimal design of periodic functionally graded composites with prescribed properties

The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.     

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.     

A review of optimization of cast parts using topology optimization

This is the second part of a two-paper review of optimization of cast parts. In the first paper, we focused on the application of the original topology optimization codes. The problems with the interpretation has been shown and discussed. In this paper, we introduce TopShape to overcome this lack. It is able to take manufacturing constraints for cast parts into account. The features of TopShape and its results will be discussed and compared with the result of the commercial code OptiStruct. Furthermore, a new design alternative for cast parts will be derived.     

Deriving the effective ultrasound equations for soft tissue interrogation

We are interested in the ultrasound interrogation of soft tissue. As soft tissue is a complicated material, we derive the effective acoustic equations using an averaging method known as homogenization to arrive at suitable constitutive equations. Soft tissue is composed of actin, elastin, collagen, and interstitial fluid. Collagen is packaged in parallel-fibered bundles, or fasciles, in the case of tendon and ligament and a meshwork of fibrils in the case of skin. The interstitial fluid is a hydrophillic gel. The purpose of this paper is to show that one may derive a general form for the constitutive equations for the aggregate soft tissue structure. Subsequent research will be directed on the estimation of the various coefficients and kernels appearing in the effective law.     

Sensitivity of optimal shapes of artificial grafts with respect to flow parameters

The difficulties arising in the numerical solution of PDE-constrained shape optimization problems are manifold. Key ingredients are the optimization strategy and the shape deformation method. Furthermore, the robustness of the optimal shape with respect to simulation parameters is of great interest. In this paper, we consider fluid flows described by the incompressible Navier–Stokes equations. Previous studies on artificial bypass grafts indicated the need for specific constitutive models to account for the non-Newtonian nature of blood; in particular, the constitutive model was shown to affect the solution of the shape optimization problem. We employ a shape optimization framework that couples a finite element solver with quasi-Newton-type optimizers and a Bezier spline shape parametrization. To compute derivatives of the optimal shapes with respect to viscosity, we transform the entire optimization framework by combining the automatic differentiation tools Adifor2 and TAPENADE. We demonstrate the impact of the geometry parametrization and of geometric constraints on the optimization outcome. Finally, we employ the transformed framework to compute the sensitivity of the optimal shape of bypass grafts with respect to kinematic viscosity. The resulting sensitivities predict very accurately the influence of viscosity changes on the optimal shape. The proposed methodology provides a powerful tool to further investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters.     

Structural optimization under uncertain loads and nodal locations

This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem.     

Interior point multigrid methods for topology optimization

In this paper, a new multigrid interior point approach to topology optimization problems in the context of the homogenization method is presented. The key observation is that nonlinear interior point methods lead to linear-quadratic subproblems with structures that can be favourably exploited within multigrid methods. Primal as well as primal-dual formulations are discussed. The multigrid approach is based on the transformed smoother paradigm. Numerical results for an example problem are presented. Received February 15, 1999     

Numerical results on the homogenization of stokes and navier-stokes equations modeling a class of problems from fluid mechanics

The aim of this paper is to present a numerical study of the homogenization theory applied to Stokes and Navier-Stokes equations with nonstandard conditions on the boundary of a periodically multiperforated domain. From a practical point of view our purpose is the numerical solution of the physical problem of the steam-water condensation in a condenser which contains a periodical structure of pipes. For this problem, we first propose a mathematical model based on theoretical results about the homogenization of Navier-Stokes equations, and then we present a numerical study of this model which allows the effective computation of the homogenized solution.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty

Aim of this work is the synthesis of auxetic structures using a topology optimization approach for micropolar (or Cosserat) materials. A distributed compliant mechanism design problem is formulated, adopting a SIMP–like model to approximate the constitutive parameters of 2D micropolar bodies. The robustness of the proposed approach is assessed through numerical examples concerning the optimal design of structures that can expand perpendicularly to an applied tensile stress. The influence of the material characteristic length on the optimal layouts is investigated. Depending on the inherent flexural stiffness of micropolar solids, truss–like solutions typical of Cauchy solids are replaced by curved beam–like material distributions. No homogenization technique is implemented, since the proposed design approach applies to elements made of microstructured material with prescribed properties and not to the material itself.     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

WYPiWYG Damage Mechanics for Soft Materials: A Data-Driven Approach

The conservative elastic behavior of soft materials is characterized by a stored energy function which shape is usually specified a priori, except for some material parameters. There are hundreds of proposed stored energies in the literature for different materials. The stored energy function may change under loading due to damage effects, but it may be considered constant during unloading–reloading. The two dominant approaches in the literature to model this damage effect are based either on the Continuum Damage Mechanics framework or on the Pseudoelasticity framework. In both cases, additional assumed evolution functions, with their associated material parameters, are proposed. These proposals are semi-inverse, semi-analytical, model-driven and data-adjusted ones. We propose an alternative which may be considered a non-inverse, numerical, model-free, data-driven, approach. We call this approach WYPiWYG constitutive modeling. We do not assume global functions nor material parameters, but just solve numerically the differential equations of a set of tests that completely define the behavior of the solid under the given assumptions. In this work we extend the approach to model isotropic and anisotropic damage in soft materials. We obtain numerically the damage evolution from experimental tests. The theory can be used for both hard and soft materials, and the infinitesimal formulation is naturally recovered for infinitesimal strains. In fact, we motivate the formulation in a one-dimensional infinitesimal framework and we show that the concepts are immediately applicable to soft materials.     

Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity

This paper presents a bidirectional evolutionary structural optimization (BESO) method for designing periodic microstructures of two-phase composites with extremal electromagnetic permeability and permittivity. The effective permeability and effective permittivity of the composite are obtained by applying the homogenization technique to the representative periodic base cell (PBC). Single or multiple objectives are defined to maximize or minimize the electromagnetic properties separately or simultaneously. The sensitivity analysis of the objective function is conducted using the adjoint method. Based on the established sensitivity number, BESO gradually evolves the topology of the PBC to an optimum. Numerical examples demonstrate that the electromagnetic properties of the resulting 2D and 3D microstructures are very close to the theoretical Hashin-Shtrikman (HS) bounds. The proposed BESO algorithm is computationally efficient as the solution usually converges in less than 50 iterations. The proposed BESO method can be implemented easily as a post-processor to standard commercial finite element analysis software packages, e.g. ANSYS which has been used in this study. The resulting topologies are clear black-and-white solutions (with no grey areas). Some interesting topological patterns such as Vigdergauz-type structure and Schwarz primitive structure have been found which will be useful for the design of electromagnetic materials.     

Topology optimization of channel flow problems

This paper describes a topology design method for simple two-dimensional flow problems. We consider steady, incompressible laminar viscous flows at low-to-moderate Reynolds numbers. This makes the flow problem nonlinear and hence a nontrivial extension of the work of Borrvall and Petersson (2003).Further, the inclusion of inertia effects significantly alters the physics, enabling solutions of new classes of optimization problems, such as velocity-driven switches, that are not addressed by the earlier method. Specifically, we determine optimal layouts of channel flows that extremize a cost function which measures either some local aspect of the velocity field or a global quantity, such as the rate of energy dissipation. We use the finite element method to model the flow, and we solve the optimization problem with a gradient-based math-programming algorithm that is driven by analytical sensitivities. Our target application is optimal layout design of channels in fluid network systems. Using concepts borrowed from topology optimization of compliant mechanisms in solid mechanics, we introduce a method for the synthesis of fluidic components, such as switches, diodes, etc.     

Developments in structural-acoustic optimization for passive noise control

Summary  Low noise constructions receive more and more attention in highly industrialized countries. Consequently, decrease of noise radiation challenges a growing community of engineers. One of the most efficient techniques for finding quiet structures consists in numerical optimization. Herein, we consider structural-acoustic optimization understood as an (iterative) minimum search of a specified objective (or cost) function by modifying certain design variables. Obviously, a coupled problem must be solved to evaluate the objective function. In this paper, we will start with a review of structural and acoustic analysis techniques using numerical methods like the finite- and/or the boundary-element method. This is followed by a survey of techniques for structural-acoustic coupling. We will then discuss objective functions. Often, the average sound pressure at one or a few points in a frequency interval accounts for the objective function for interior problems, wheareas the average sound power is mostly used for external problems. The analysis part will be completed by review of sensitivity analysis and special techniques. We will then discuss applications of structural-acoustic optimization. Starting with a review of related work in pure structural optimization and in pure acoustic optimization, we will categorize the problems of optimization in structural acoustics. A suitable distinction consists in academic and more applied examples. Academic examples iclude simple structures like beams, rectangular or circular plates and boxes; real industrial applications consider problems like that of a fuselage, bells, loudspeaker diaphragms and components of vehicle structures. Various different types of variables are used as design parameters. Quite often, locally defined plate or shell thickness or discrete point masses are chosen. Furthermore, all kinds of structural material parameters, beam cross sections, spring characteristics and shell geometry account for suitable design modifications. This is followed by a listing of constraints that have been applied. After that, we will discuss strategies of optimization. Starting with a formulation of the optimization problem we review aspects of multiobjective optimization, approximation concepts and optimization methods in general. In a final chapter, results are categorized and discussed. Very often, quite large decreases of noise radiation have been reported. However, even small gains should be highly appreciated in some cases of certain support conditions, complexity of simulation, model and large frequency ranges. Optimization outcomes are categorized with respect to objective functions, optimization methods, variables and groups of problems, the latter with particular focus on industrial applications. More specifically, a close-up look at vehicle panel shell geometry optimization is presented. Review of results is completed with a section on experimental validation of optimization gains. The conclusions bring together a number of open problems in the field.