Design of fracture resistant energy absorbing structures using elastoplastic topology optimization

This paper introduces a novel elastoplastic topology optimization method for fracture resistant energy absorbing structural designs. The target is to find the optimal structural topologies with high plastic work absorption capacity while constraining the fracture indicators below the prescribed constraints. As the two main fracture mechanisms in ductile metals, the ductile fracture and shear fracture criteria are considered using uncoupled CrachFEM fracture model. A consistent adjoint method is presented for the path-dependent sensitivity analysis under the plane stress assumption. Several numerical examples are carried out to demonstrate the effectiveness of the proposed method. It is shown that by constraining the fracture indicators, the optimized designs have a more uniform plastic work distribution and high ductility with a significant delay of failure points. This eventually leads to much better material utilization with enhanced ultimate energy absorption capacities.     

Continuous transportation as a material distribution topology optimization problem

The problem of moving a commodity with a given initial mass distribution to a pre-specified target mass distribution so that the total work is minimized can be traced back at least to Monge’s work from 1781. Here, we consider a version of this problem aiming to minimize a combination of road construction and transportation cost by determining, at each point, the local direction of transportation. This paper covers the modeling of the problem, highlights how it can be formulated as a material distribution topology optimization problem, and shows some results.

     

Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.     

Robust topology optimization accounting for misplacement of material

The use of topology optimization for structural design often leads to slender structures. Slender structures are sensitive to geometric imperfections such as the misplacement or misalignment of material. The present paper therefore proposes a robust approach to topology optimization taking into account this type of geometric imperfections. A density filter based approach is followed, and translations of material are obtained by adding a small perturbation to the center of the filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics during the optimization process. The proposed method is successfully applied to three example problems: the minimum compliance design of a slender column-like structure and a cantilever beam and a compliant mechanism design. An extensive Monte Carlo simulation is used to show that the obtained topologies are more robust with respect to geometric imperfections.     

Designing meta material slabs exhibiting negative refraction using topology optimization

This paper proposes a topology optimization based approach for designing meta materials exhibiting a desired negative refraction with high transmission at a given angle of incidence and frequency. The approach considers a finite slab of meta material consisting of axis-symmetric designable unit cells subjected to an exterior field. The unit cell is designed to achieve the desired properties based on tailoring the response of the meta material slab under the exterior field. The approach is directly applicable to physical problems modeled by the Helmholtz equation, such as acoustic, elastic and electromagnetic wave problems. Acoustic meta materials with unit cell size on the order of half the wave length are considered as examples. Optimized designs are presented and their performance under varying frequency and angle of incidence is investigated.     

Viscoelastic material design with negative stiffness components using topology optimization

An application of topology optimization to design viscoelastic composite materials with elastic moduli that soften with frequency is presented. The material is a two-phase composite whose first constituent is isotropic and viscoelastic while the other is an orthotropic material with negative stiffness but stable. A concept for this material based on a lumped parameter model is used. The performance of the topology optimization approach in this context is illustrated using three examples.     

Design of buckling-induced mechanical metamaterials for energy absorption using topology optimization

A novel design concept for buckling-induced mechanical metamaterials for energy absorption is presented. The force-displacement curves of the mechanical metamaterials are analyzed according to the curves of their unit cells, and the energy-absorbing characteristics of mechanical metamaterials are evaluated. Two topology optimization models are proposed. One maximizes the buckling-induced dissipated energy to facilitate the design of metamaterials with high energy absorption and low elastic strain energy. The other maximizes the dissipated energy with a constraint that the mechanical metamaterials should be self-recoverable. An energy interpolation scheme is employed to avoid numerical instabilities in the geometric nonlinear finite element analysis. A two-phase algorithm is proposed to find the optimized result from a uniform initial guess, and sensitivity analysis is performed. The optimized design has a larger amount of buckling-induced dissipated energy than the previously proposed structural prototypes. Moreover, the self-recoverable mechanical metamaterial is successfully designed by topology optimization.     

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.     

Stress-based topology optimization of compliant mechanisms design using geometrical and material nonlinearities

Structural and Multidisciplinary Optimization - In this work, a density-based method is applied for synthesizing compliant mechanisms using topology optimization. This kind of mechanisms uses the...     

Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization

This paper presents a method to locally constrain multiple material volume domains for structural optimization with the Level Set Method (LSM). Two different Lagrangian formulations and multiplier update methods are used, for both the global and local problem. The local volume domains can be constrained by both equality and inequality constraints. The optimization objective is compliance minimization for well-posed statically loaded structures. For validation, several example problems are established and solved using the proposed method. Results show that the volume ratios for user established sub-domains can be controlled successfully. The local constraint values are met accurately in the case of equality constraints and remain in their feasible domain in the case of inequality constraints. Optimization results are not significantly hindered by the introduction of local volume constraints for comparable problems.     

Topology optimization of sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems

Due to the significant difference of dynamic properties between the fluid medium and the structure, when a vibro-acoustic system is subjected to a higher frequency excitation, it may typically exhibit mid-frequency behavior which involves different wavelength deformations and is very sensitive to the uncertainties of the system. This paper deals with optimized distribution of a sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems by using hybrid boundary element analysis and statistical energy analysis. Based on the solid isotropic material with penalization approach, an artificial sound absorbing material model is suggested and the relative densities of the sound absorbing material are taken as design variables. The sound pressure level at a specified point in the acoustic cavity is to be minimized by distributing a given amount of sound absorbing material. An efficient direct differentiation scheme for the response sensitivity analysis is proposed. Then, the optimization problem is solved by using the method of moving asymptotes. A numerical example illustrates the validity and effectiveness of the present optimization model. Impact of the excitation frequency on optimized topology is also discussed.

     

Microstructural topology optimization with respect to sound power radiation

The paper deals with the problem of topological design of microstructure with respect to minimization of the sound power radiation from a vibrating macrostructure. The macrostructure is excited at a single or a band of excitation frequencies by a time-harmonic mechanical loading with prescribed amplitude and spatial distribution. The structural damping is considered to be proportional damping. The sound power is calculated using a high frequency approximation formulation and thus the sensitivity analysis may be performed in a very efficient manner. The microstructure composed of two different solid isotropic materials is assumed to be identical from point to point at the macro-level which implies that the interface between the structure and the acoustic medium is unchanged during the design process. The equivalent material properties of the macrostructure are calculated using homogenization method and the bi-material SIMP model is employed to achieve zero-one design at the micro-scale. Numerical examples are given to validate the model developed. Some interesting features of acoustic microstructure topology optimization are revealed and discussed.     

Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.     

Simultaneous parametric material and topology optimization with constrained material grading

We consider the problem of parametric material and simultaneous topology optimization of an elastic continuum. To ensure existence of solutions to the proposed optimization problem and to enable the imposition of a deliberate maximal material grading, two approaches are adopted and combined. The first imposes pointwise bounds on design variable gradients, whilst the second applies a filtering technique based on a convolution product. For the topology optimization, the parametrized material is multiplied with a penalized continuous density variable. We suggest a finite element discretization of the problem and provide a proof of convergence for the finite element solutions to solutions of the continuous problem. The convergence proof also implies the absence of checkerboards. The concepts are demonstrated by means of numerical examples using a number of different material parametrizations and comparing the results to global lower bounds.     

Design of phononic crystals for self-collimation of elastic waves using topology optimization method

Self-collimating phononic crystals (PCs) are periodic structures that enable self-collimation of waves. While various design parameters such as material property, period, lattice symmetry, and material distribution in a unit cell affect wave scattering inside a PC, this work aims to find an optimal material distribution in a unit cell that exhibits the desired self-collimation properties. While earlier studies were mainly focused on the arrangement of self-collimating PCs or shape changes of inclusions in a unit cell having a specific topological layout, we present a topology optimization formulation to find a desired material distribution. Specifically, a finite element based formulation is set up to find the matrix and inclusion material distribution that can make elastic shear-horizontal bulk waves propagate along a desired target direction. The proposed topology optimization formulation newly employs the geometric properties of equi-frequency contours (EFCs) in the wave vector space as essential elements in forming objective and constraint functions. The sensitivities of these functions with respect to design variables are explicitly derived to utilize a gradient-based optimizer. To show the effectiveness of the formulation, several case studies are considered.     

Design of phononic materials/structures for surface wave devices using topology optimization

We develop a topology optimization approach to design two- and three-dimensional phononic (elastic) materials, focusing primarily on surface wave filters and waveguides. These utilize propagation modes that transmit elastic waves where the energy is contained near a free surface of a material. The design of surface wave devices is particularly attractive given recent advances in nano- and micromanufacturing processes, such as thin-film deposition, etching, and lithography, which make it possible to precisely place thin film materials on a substrate with submicron feature resolution. We apply our topology optimization approach to a series of three problems where the layout of two materials (silicon and aluminum) is sought to achieve a prescribed objective: (1) a grating to filter bulk waves of a prescribed frequency in two and three dimensions, (2) a surface wave device that uses a patterned thin film to filter waves of a single or range of frequencies, and (3) a fully three-dimensional structure to guide a wave generated by a harmonic input on a free surface to a specified output port on the surface. From the first to the third example, the resulting topologies increase in sophistication. The results demonstrate the power and promise of our computational framework to design sophisticated surface wave devices.     

Design of thermoelastic multi-material structures with graded interfaces using topology optimization

A level-set based shape and topology optimization framework is used to investigate the influence of graded interfaces in the optimization of micro-architectured multi-materials. In contrast to other studies found in the literature, interfaces are considered as smooth and graded transitions between constitutive phases instead of sharp delimitations. Case studies for extreme thermoelastic properties of 2D isotropic composites are analyzed and optimal designs are presented. It is shown that explicitly accounting for interfaces can influence the design of heterogeneous materials in composite microstructures.     

Design of conjugate, conjoined shapes and tilings using topology optimization

We present a method for obtaining conjugate, conjoined shapes and tilings in the context of the design of structures using topology optimization. Optimal material distribution is achieved in topology optimization by setting up a selection field in the design domain to determine the presence/absence of material there. We generalize this approach in this paper by presenting a paradigm in which the material left out by the selection field is also utilised. We obtain conjugate shapes when the region chosen and the region left-out are solutions for two problems, each with a different functionality. On the other hand, if the left-out region is connected to the selected region in some pre-determined fashion for achieving a single functionality, then we get conjoined shapes. The utilization of the left-out material, gives the notion of material economy in both cases. Thus, material wastage is avoided in the practical realization of these designs using many manufacturing techniques. This is in contrast to the wastage of left-out material during manufacture of traditional topology-optimized designs. We illustrate such shapes in the case of stiff structures and compliant mechanisms. When such designs are suitably made on domains of the unit cell of a tiling, this leads to the formation of new tilings which are functionally useful. Such shapes are not only useful for their functionality and economy of material and manufacturing, but also for their aesthetic value.