Perimeter Penalization Research Articles

Phase Field Model for Multi-Material Shape Optimization of Inextensible Rods

We derive a model for the optimization of the bending and torsional rigidities of nonhomogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via Γ-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.

Regularity results for an optimal design problem with lower order terms

AbstractWe study the regularity of the interface for optimal energy configurations of functionals involving bulk energies with an additional perimeter penalization of the interface. Here we allow a more general structure for the energy functional in the bulk term. For a minimal configuration(E,u){(E,u)}, the Hölder continuity ofuis well known. We give an estimate for the singular set of the boundary∂⁡E{\partial E}. Namely we show that the Hausdorff dimension of the singular set is strictly smaller thann-1{n-1}.

A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is 1∕2 and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

Multi-material topology optimization of compliant mechanisms using regularized projected gradient approach

In this article, the identification of optimal topologies for multi-material compliant mechanisms based on the regularized projected gradient approach is presented and analyzed. The monolithic structure of compliant mechanisms makes them suitable to be used in microelectromechanical systems to transfer load and force. The use of additive manufacturing technologies makes it possible to build multi-material structures. However, a proper design optimization approach must be considered to design multi-material mechanisms as well. In this article, the solid isotropic material with a penalization (SIMP) material interpolation scheme is used to parameterize the continuum design domain. The perimeter penalization approach is utilized to remove the checkerboard patterns and scatters in optimal designs. To handle the volume equality constraint, the regularized constraint projection strategy is utilized. Some benchmark problems are considered to analyze the performance of the method.

Binary recovery via phase field regularization for first-arrival traveltime tomography

We propose a double obstacle phase field methodology for binary recovery of the slowness function of an Eikonal equation found in first-arrival traveltime tomography. We treat the inverse problem as an optimization problem with quadratic misfit functional added to a phase field relaxation of the perimeter penalization functional. Our approach yields solutions as we account for well posedness of the forward problem by choosing regular priors. We obtain a convergent finite difference and mixed finite element based discretization and a well defined descent scheme by accounting for the non-differentiability of the forward problem. We validate the phase field technique with a —convergence result and numerically by conducting parameter studies for the scheme, and by applying it to a variety of test problems with different geometries, boundary conditions, and source—receiver locations.

Relaxation for Optimal Design Problems with Non-standard Growth

In this paper we investigate the possibility of obtaining a measure representation for functionals arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided.

Regularity for free interface variational problems in a general class of gradients

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$ (u,A) \quad \mapsto \quad \int_\Omega 2fu \; \text{d}x \; - \int_{\Omega \cap A} \sigma_1 \mathscr A u \cdot \mathscr A u \; \text{d}x \; - \int_{\Omega \setminus A} \sigma_2\mathscr A u\cdot \mathscr A u \; \text{d}x \; + \; \text{Per}(A;\overline \Omega),$$ where $\Omega$ is a bounded Lipschitz domain, $A\subset \mathbb R^N$ is a Borel set, $u:\Omega \subset \mathbb R^N \to \mathbb R^d$, $\mathscr A$ is an operator of gradient form, and $\sigma_1, \sigma_2$ are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on $f$, we show for a solution $(w,A)$, that the topological boundary of $A \cap \Omega$ is locally a $\rm{C}^1$-hypersurface up to a closed set of zero $\mathscr H^{N-1}$-measure.

Shape optimization for surface functionals in Navier-Stokes flow using a phase field approach

We consider shape and topology optimization of an object in fluid flow governed by the Navier–Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg–Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn–Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.

Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints

The wear of materials continues to be a limiting factor in the lifetime and performance of mechanical systems with sliding surfaces. As the demand for low wear materials grows so does the need for models and methods to systematically optimize tribological systems. Elastic foundation models offer a simplified framework to study the wear of multimaterial composites subject to abrasive sliding. Previously, the evolving wear profile has been shown to converge to a steady-state that is characterized by a time-independent elliptic equation. In this article, the steady-state formulation is generalized and integrated with shape optimization to improve the wear performance of bi-material composites. Both macroscopic structures and periodic material microstructures are considered. Several common tribological objectives for systems undergoing wear are identified and mathematically formalized with shape derivatives. These include (i) achieving a planar wear surface from multimaterial composites and (ii) minimizing the run-in volume of material lost before steady-state wear is achieved. A level-set based topology optimization algorithm that incorporates a novel constraint on the level-set function is presented. In particular, a new scheme is developed to update material interfaces; the scheme (i) conveniently enforces volume constraints at each iteration, (ii) controls the complexity of design features using perimeter penalization, and (iii) nucleates holes or inclusions with the topological gradient. The broad applicability of the proposed formulation for problems beyond wear is discussed, especially for problems where convenient control of the complexity of geometric features is desired.

Optimal Fine‐Scale Structures in Compliance Minimization for a Shear Load

AbstractWe return to a classic problem of structural optimization whose solution requires microstructure. It is well‐known that perimeter penalization assures the existence of an optimal design. We are interested in the regime where the perimeter penalization is weak; i.e., in the effect of perimeter as a selection mechanism in structural optimization. To explore this topic in a simple yet challenging example, we focus on a two‐dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear. We consider the minimization of a weighted sum of volume, perimeter, and compliance (i.e., the work done by the load), focusing on the behavior as the weight ɛ of the perimeter term tends to 0. Our main result concerns the scaling of the optimal value with respect to ɛ. Our analysis combines an upper bound and a lower bound. The upper bound is proved by finding a near‐optimal structure, which resembles a rank‐2 laminate except that the approximate interfaces are replaced by branching constructions. The lower bound, which shows that no other microstructure can be much better, uses arguments based on the Hashin‐Shtrikman variational principle. The regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures. While perimeter has been considered as a selection mechanism in other problems involving microstructure, the example considered here is novel because optimality seems to require the use of two well‐separated length scales.© 2016 Wiley Periodicals, Inc.

On the regularity of critical and minimal sets of a free interface problem

We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin [15].

Relaxation for an optimal design problem with linear growth and perimeter penalization

This paper is devoted to the relaxation and integral representation in the space of functions of bounded variation for an integral energy arising from optimal design problems. The presence of a perimeter penalization is also considered in order to avoid non-existence of admissible solutions and, in addition, this leads to an interaction in the limit energy. More general models have also been taken into account.

More counterexamples to regularity for minimizers of the average-distance problem

Abstract The average-distance problem, in the penalized formulation, involves minimizing (1) E μ λ(Σ) := ∫ℝ d d(x,Σ)dμ(x) + λℋ 1 (Σ), among path-wise connected, closed sets Σ with finite ℋ 1 -measure, where d ≥ 2, μ is a given measure, λ is a given parameter and d(x,Σ) := inf y∈Σ|x - y|. The average-distance problem can be also considered among compact, convex sets with perimeter and/or volume penalization, i.e. minimizing (2) ℰ(μ,λ1,λ2)(·):=∫ℝ d d(x,·)dμ(x) + λ1Per(·) + λ2Vol(·), where μ is a given measure, λ1,λ2 ≥ 0 are given parameters with λ1 + λ2 > 0, and the unknown varies among compact, convex sets. Very little is known about the regularity of minimizers of (2). In particular it is unclear if minimizers of (2) are in general C 1 regular. The aim of this paper is twofold: first, we provide in ℝ2 a second approach in constructing minimizers of (1) which are not C 1 regular; then, using the same technique, we provide an example of minimizer of (2) whose border is not C 1 regular, under perimeter penalization only.

3D–2D dimensional reduction for a nonlinear optimal design problem with perimeter penalization

A 3D–2D dimension reduction for a nonlinear optimal design problem with a perimeter penalization is performed in the realm of Γ-convergence, providing an integral representation for the limit functional.

Topology optimization methods with gradient-free perimeter approximation

In this paper we introduce a family of smooth perimeter approximating functionals designed to be incorporated within topology optimization algorithms. The required mathematical properties, namely the \Gamma -convergence and the compactness of sequences of minimizers, are first established. Then we propose several methods for the solution of topology optimization problems with perimeter penalization showing different features. We conclude by some numerical illustrations in the contexts of least square problems and compliance minimization.

A Nonlocal Mean Curvature Flow and Its Semi-implicit Time-Discrete Approximation

We study in this paper the geometric evolution of a set $E$, with a velocity given by a “curvature” of $\partial E$ which is nonlocal and singular at the origin, in the sense that it behaves like a power of the classical curvature. This curvature is the first variation of an energy which is proportional to the volume of the set of points at a given distance to $\partial E$, and which was proposed in a recent work of Barchiesi et al. [Multiscale Model. Simul., 8 (2010), pp. 1715--1741] as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with the degeneracies of our problem, we first give an abstract existence and uniqueness result for viscosity solutions of nonlocal degenerate Hamiltonians, satisfying a suitable continuity assumption with respect to Kuratowski convergence of the level sets. This abstract setting applies to an approximated variant of our flow. Then, by the method of minimizing movements, we also build a weak solution of our curvature flow. We illustrate this with some examples and compare the results with the standard mean curvature flow.

Reconstruction of material losses by perimeter penalization and phase-field methods

We treat the inverse problem of determining material losses, such as cavities, in a conducting body, by performing electrostatic measurements at the boundary. We develop a numerical approach, based on variational methods, to reconstruct the unknown material loss by a single boundary measurement of current and voltage type. The method is based on the use of phase-field functions to model the material losses and on a perimeter-like penalization to regularize the otherwise ill-posed problem. We justify the proposed approach by a convergence result, as the error on the measurement goes to zero.

Regularization of ill-posed Mumford–Shah models with perimeter penalization

The problems of existence, stability and convergence of minimizers of a Mumford–Shah functional for the simultaneous reconstruction and segmentation of a distributed parameter in an ill-posed operator equation are considered. An appropriate set of regular partitions of the domain of the distributed parameter is introduced and a distance for partitions is constructed. Pairs of partitions and coefficient vectors form the space of feasible variables for the Mumford–Shah functional. Each such pair defines a piecewise constant function. The relation between convergence of geometric and vectorial variables on the one hand and convergence of the associated piecewise constant functions in L1 on the other is investigated under the assumption of injectivity of the operator equation on the set of piecewise constant parameters. Compactness and semi-continuity results for regular partitions are presented and used to prove existence, stability with respect to perturbations in the data and convergence of the solution if the error level goes to zero. Numerical experiments in a computerized tomography setting demonstrate the validity of the derived convergence result.

Regularity of components in optimal design problems with perimeter penalization

We consider minimal energy configurations of mixtures of two materials in $\Omega\subset\mathbb{R}^2$ , where the energy includes a penalty on the length of the interface between the materials. We show that, for one of the materials, the boundary of each component is smooth, and we prove the existence of an upper bound on the relative distances between components.

An optimal design problem with perimeter penalization

We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.