Domain Perturbation Research Articles

A localized domain perturbation which splits the spectrum of the Laplacian

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The second one consists in the deformation of the boundary of the domain itself, and requires more technical properties of the bilinear form of the variational problem.

Effective permeability tensor of confined flows with wall grooves of arbitrary shape

Pressure and shear-driven flows of a confined film of fluid overlying a periodic one-dimensional topography of arbitrary shape are considered for prediction of the effective hydraulic permeability in the Stokes flow regime. The other surface confining the fluid may be a planar no-slip wall, an identically patterned wall, a free surface or a surface with prescribed shear. Analytical predictions are obtained using spectral analysis and the domain perturbation method under the assumption of small pattern size to pitch ratio. Using a novel decomposition of the channel height effects into exponentially and algebraically decaying components, a simple surface-metrology-dependent relationship which connects the eigenvalues of the effective permeability tensor is obtained. Two representative topographies are assessed numerically: the infinitely differentiable topography of a phase-modulated sinusoid which has multiple local extrema and zero crossings and the non-differentiable triangular-wave topography. Non-differentiability in the form of corners of triangular patterns and the cusps of scalloped patterns are not found to be an impediment to meaningful and numerically accurate asymptotic predictions of effective permeability and effective slip, contradicting an earlier suggestion from the literature. Several distinct applications of the theory to the friction-reduction and shear-stability performance of the recently developed lubricant impregnated patterned surfaces as well as to scalloped and trapezoidal drag-reduction riblets are discussed, with comparison to experimental data from the literature for the last application. Analytical approximations which have an extended domain of numerical accuracy are also proposed.

Analyticity of Steklov eigenvalues of nearly circular and nearly spherical domains

We consider the Dirichlet-to-Neumann operator (DNO) on nearly circular and nearly spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (J Comput Phys 194(1):278–303, 2004. https://doi.org/10.1016/j.jcp.2003.09.006); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to-Neumann operator.

Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Pointwise error analysis of the linear finite element approximation for $$-\,\Delta u + u = f$$ in $$\Omega $$, $$\partial _n u = \tau $$ on $$\partial \Omega $$, where $$\Omega $$ is a bounded smooth domain in $$\mathbb {R}^N$$, is presented. We establish $$O(h^2|\log h|)$$ and O(h) error bounds in the $$L^\infty $$- and $$W^{1,\infty }$$-norms respectively, by adopting the technique of regularized Green’s functions combined with local $$H^1$$- and $$L^2$$-estimates in dyadic annuli. Since the computational domain $$\Omega _h$$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $$\Omega _h \ne \Omega $$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

On a Babuška Paradox for Polyharmonic Operators: Spectral Stability and Boundary Homogenization for Intermediate Problems

We analyse the spectral convergence of high order elliptic differential operators subject to singular domain perturbations and homogeneous boundary conditions of intermediate type. We identify sharp assumptions on the domain perturbations improving, in the case of polyharmonic operators of higher order, conditions known to be sharp in the case of fourth order operators. The optimality is proved by analysing in detail a boundary homogenization problem, which provides a smooth version of a polyharmonic Babu\v{s}ka paradox.

Optimization of the electromagnetic scattering problem based on the topological derivative method.

A new optimization method based on the topological derivative concept is developed for the electromagnetic design problem. Essentially, the purpose of the topological derivative method is to measure the sensitivity of a given shape functional with respect to a singular domain perturbation, so that it has applications in many relevant fields such as shape and topology optimization for imaging processing, inverse problems, and design of metamaterials. The topological derivative is rigorously derived for the electromagnetic scattering problem and used as gradient descent direction to find local optima for the design of electromagnetic devices. We demonstrate that the resulting topology design algorithm is remarkably simple and efficient and naturally leads to binary designs, while depending only on the solution of the conventional finite element formulation for electrodynamics. Finally, several numerical experiments in two and three spatial dimensions are presented to illustrate the performance of the proposed formulation.

On qualitative properties of solutions for elliptic problems with the p-Laplacian through domain perturbations

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem in a bounded domain upon domain perturbations. Assuming that the nonlinearity f is superlinear and subcritical, we establish Hadamard-type formulas for such critical levels. As an application, we show that among all (generally eccentric) spherical annuli Ω least nontrivial critical levels attain maximum if and only if Ω is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions whenever Ω is a ball or concentric annulus.

On the (in)stability of the identity map in optimal transportation

We collect some examples of optimal transports for quadratic cost in order to explore the stability of the identity map as an optimal transport. First, we consider density and domain perturbations near regular portions of domains. Second, we investigate density and domains deformations to non-regular parts of domains. Here, we restrict our attention to two dimensions and focus near 90 $$^{\circ }$$ corners.

Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domainΩ⊂ ℝN(N= 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary conditionu · n∂Ω = gon∂Ω. Because the original domainΩmust be approximated by a polygonal (or polyhedral) domainΩhbefore applying the finite element method, we need to take into account the errors owing to the discrepancyΩ ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation ofn∂Ωbyn∂Ωhmakes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operatorH1(Ω)N→H1/2(∂Ω);u↦u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimatesO(hα+ε) andO(h2α+ε) for the velocity in theH1- andL2-norms respectively, whereα = 1 ifN = 2 andα = 1/2 ifN = 3. This improves the previous result [T. Kashiwabaraet al.,Numer. Math.134(2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameterϵin the estimates.

Behaviour of the Stokes operators under domain perturbation

Depending on the geometry of the domain, one can define—at least—three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.

Improved asymptotic predictions for the effective slip over a corrugated topography

• Significantly improved asymptotic predictions on effective slip in comparison to literature. • Systematic study on convergence radii of asymptotic series. • Novel use of series improvement and acceleration techniques. • Novel use of spectral analysis and symbolic computation. Accurate analytical prediction of the effective slip boundary condition in shear-driven Stokes flows directed longitudinally and transversely to a one-dimensional sinusoidal no-slip topography is explored. First, the domain perturbation technique is extended through spectral analysis and symbolic computations to obtain polynomial approximations (Taylor polynomials) of arbitrary specifiable order for the effective slip length. However, when assessed for numerical accuracy against fully-resolved numerical simulations using the finite-element-method, higher order Taylor polynomials give progressively inferior predictions in comparison to lower-order ones, unless the product of amplitude and wave-number is restricted below unity. From Domb–Sykes plots, the reason for the poor accuracy of higher order Taylor polynomials is assessed to be the finite convergence radii, approximately equaling unity, of the asymptotic power series for both longitudinal and transverse flows. For either of the flows, application of Euler transformation to the expansion parameter provide polynomial-form approximations that are accurate for amplitude values exceeding the convergence radius. The slow convergence of the Euler-transformed series can be remedied through Shanks transformation, at the cost of losing the convenience of closed forms. Finally, Padé approximants are shown to provide even more accurate but still closed-form alternatives to polynomials that work accurately at amplitudes much exceeding the above-identified convergence radii.

Spectral stability for a class of fourth order Steklov problems under domain perturbations

We study the spectral stability of two fourth order Steklov problems upon domain perturbation. One of the two problems is the classical DBS - Dirichlet Biharmonic Steklov - problem, the other one is a variant. Under a comparatively weak condition on the convergence of the domains, we prove the stability of the resolvent operators for both problems, which implies the stability of eigenvalues and eigenfunctions. The stability estimates for the eigenfunctions are expressed in terms of the strong $H^2$-norms. The analysis is carried out without assuming that the domains are star-shaped. Our condition turns out to be sharp at least for the variant of the DBS problem. In the case of the DBS problem, we prove stability of a suitable Dirichlet-to-Neumann type map under very weak conditions on the convergence of the domains and we formulate an open problem. As bypass product of our analysis, we provide some stability and instability results for Navier and Navier-type boundary value problems for the biharmonic operator.

Steklov eigenvalues of reflection-symmetric nearly circular planar domains.

We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

Critical Domains for the First Nonzero Neumann Eigenvalue in Riemannian Manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace–Beltrami operator on bounded subdomains $$\Omega $$ of a Riemannian manifold $$({\mathscr {M}},g)$$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $$\mu _2(\Omega )$$ with respect to volume-preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger’s shape derivative formula which covers the case when $$\mu _2(\Omega )$$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $${\mathscr {M}}= {\mathbb {R}}^k \times {\mathscr {N}}$$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.

Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.

Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, source terms and cracks. This new concept in shape sensitivity analysis generalizes the shape derivatives from the domain boundary to its interior for admissible domains in two and three spatial dimensions. Therefore, the concept of topological derivative is a powerful tool for solving shape–topology optimization problems. There are now applications of topological derivative in many different fields of engineering and physics, such as shape and topology optimization in structural mechanics, inverse problems for partial differential equations, image processing, multiscale material design and mechanical modeling including damage and fracture evolution phenomena. In this second part of the review, a topology optimization algorithm based on first-order topological derivatives is presented. The appropriate level-set domain representation method is employed within the iterations in order to design an optimal shape–topology local solution. The algorithm is successfully used for numerical solution of a wide class of shape–topology optimization problems.

Probing Gallate-Mediated Selectivity and High-Affinity Binding of Epigallocatechin Gallate: a Way-Forward in the Design of Selective Inhibitors for Anti-apoptotic Bcl-2 Proteins.

Selective inhibition is a key focus in the design of chemotherapeutic compounds that can abrogate the oncogenic activities of anti-apoptotic Bcl-2 proteins. Although recent efforts have led to the development of highly selective BH3 mimetics, setbacks such as toxicities have limited their use in cancer therapy. Epigallocatechingallate (EGCG) has been widely reported to selectively inhibit Bcl-2 and Bcl-xL compared to other green tea phenols due to its gallate group. Herein, we investigate the interaction dynamics of EGCG at the hydrophobic grooves of Bcl-2 and Bcl-xL and the consequential effects on their BH4 domains. Arg143 and Asp108 (Bcl-2), and Glu96 and Tyr195 (Bcl-xL) formed high-affinity hydrogen interactions with the gallate group while non-gallate groups of EGCG formed weak interactions. EGCG-bound proteins showed systemic perturbations of BH4 domains coupled with the burial of crucial surface-exposed residues such as Lys17 (Bcl-2) and Asp11 (Bcl-xL); hence, a distortion of non-canonical domain interactions. Interactions of gallate group of EGCG with key hydrophobic groove residues underlie EGCG selectivity while concurrent BH4 domain perturbations potentiate EGCG inhibitory activities. Findings will aid the optimization and design of selective inhibitors that could suppress anti-apoptotic activities of Bcl2-family proteins with minimal toxicities.

Essential spectrum of a periodic waveguide with non-periodic perturbation

We consider the spectral Dirichlet–Laplacian problem on a domain which is formed from a periodic waveguide Π perturbed by non-compact, non-periodic changes of geometry. We show that the domain perturbation causes an addition to the essential spectrum, which consists of isolated points belonging to the discrete spectrum of a model problem. This model problem is posed on a domain, which is just a compact perturbation of Π. We discuss the position of the new spectral components in relation to the essential spectrum of the problem in Π.

Perturbation Sensitivity Analysis and Dynamic Topology Optimization for Heat Conduction Structure

Topology optimization is a fundamental problem for steady heat conduction. Due to the unidirectional treatment for the intermediate density elements in the traditional topology optimization, we propose a Bi-Directional Interpolation Model (BDIM) that intermediate density elements are judged by a threshold, and establish optimization model. Besides, considering the generation of new hole boundary in the process of heat conduction enhancement topology optimization, we study the perturbation sensitivity analysis with respect to new hole, and present an adaptive dynamic boundary method for the new boundary in the optimization process. Furthermore, based on the domain perturbation technique and Lebesgue differential theory, the topological derivative formulas with different objective functions subjected to three kinds of boundary conditions are derived for the control system of Poisson equation. Finally, a number of numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method for designing the heat conduction structure.

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.