On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications

We consider optimal shape design in Stokes flow using $H^1$ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary $H^1$ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed $H^1$ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.     

A gradient descent method for solving an inverse coefficient heat conduction problem

An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient. These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration convergence rates are compared for different descent parameters.     

An Improved Implementation of An Iterative Method in Boundary Identification Problems

In this paper, an inverse problem of determining geometric shape of a part of the boundary of a bounded domain is considered. Based on a conjugate gradient method, employing the adjoint equation to obtain the descent direction, an identification scheme is developed. The implementation of the method based on the boundary element method (BEM) is also discussed. This method solves the inverse boundary problem accurately without a priori information about the unknown shape to be estimated.     

Effective Shape Optimization of Laplace Eigenvalue Problems Using Domain Expressions of Eulerian Derivatives

We consider to solve numerically the shape optimization models with Dirichlet Laplace eigenvalues. Both volume-constrained and volume unconstrained formulations of the model problems are presented. Different from the literature using boundary-type Eulerian derivatives in shape gradient descent methods, we advocate to use the more general volume expressions of Eulerian derivatives. We present two shape gradient descent algorithms based on the volume expressions. Numerical examples are presented to show the more effectiveness of the algorithms than those based on the boundary expressions.     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

Strong stabilization of the system of linear elasticity by a Dirichlet boundary feedback

In this paper, by using the Nagy–Foias–Foguel theoryof decomposition of continuous semigroups of contractions, we prove that the system of linear elasticity is strongly stabilizable by a Dirichlet boundary feedback. We also give a concise proofof a theorem of Dafermos about the stability of thermoelasticity.     

Initial-boundary value problems for non-linear equations of generalized thermoelasticity and elasticity

We formulate a local existence theorem for the initial-boundary value problems of generalized thermoelasticity and classical elasticity. We present a unified approach to such boundary conditions as, for example, the boundary condition of traction, pressure or place combined with the boundary condition of heat flux or temperature.     

L ∞ stability of the MUSCL methods

We present a general L ^∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L ^∞ stability.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Exact-gradient shape optimization of a 2-D Euler flow

The optimization of an obstacle shape immersed in an Eulerian flow is investigated. In order to construct a descent method, we consider the differentiation of the flow solution with respect to the shape. In the continous case, the Hadamard variational formula yields the formal derivatives. In the discrete case, we choose an upwind method with flux splitting, and proved that an exact gradient can be derived using the adjoint state. The behavior of a gradient method is studied for a family of nozzle flows.     

Fast gradient descent method for Mean-CVaR optimization

We propose an iterative gradient descent algorithm for solving scenario-based Mean-CVaR portfolio selection problem. The algorithm is fast and does not require any LP solver. It also has efficiency advantage over the LP approach for large scenario size.     

Analysis and Application of the Rational B-Spline Finite Element Method in 2D

Non-Uniform Rational B-Splines (NURBS) are basis functions used in CAD software to describe exact geometric models. The implementation of these basis functions in the context of the Finite Element Analysis (FEA) is known as isogeometric analysis. The concept and definition of NURBS is briefly presented here. Since these functions are implemented as shape functions for the isogeometric analysis, the refinement strategies are discussed. The example of an infinite plate with circular hole serves as a benchmark. Finally, isogeometric analysis is applied to gradient elasticity since NURBS functions are of higher continuity and this is required in gradient elasticity. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type

This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.     

Fast and accurate algorithm for cavities reconstruction in an elasticity problem

The aim of this work is to reconstruct the location and geometry of a cavity embedded in a linear isotropic material Ω via an exterior boundary measurement of the displacement field. The considered problem is governed by the linear elasticity system. This inverse problem of geometry reconstruction (ie, location and shape) is formulated as a topology optimization one and solved by minimizing a Kohn‐Vogelius type functional with the help of the topological sensitivity method. Some numerical results are presented using a noniterative geometric algorithm.     

Modeling,shape analysis and computation of the equilibrium pore shape near a PEM–PEM intersection

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: (i) the existence and uniqueness of the electrical potential, (ii) the shape derivatives of state variables and (iii) the shape differentiability of the corresponding energy and the corresponding Euler–Lagrange equation. The latter leads to a modified Young–Laplace equation on the interface. This modified equation is compared with the classical Young–Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.     

Inverse thermal imaging in materials with nonlinear conductivity by material and shape derivative method

The material and shape derivative method is used for an inverse problem in thermal imaging. The goal is to identify the boundary of unknown inclusions inside an object by applying a heat source and measuring the induced temperature near the boundary of the sample. The problem is studied in the framework of quasilinear elliptic equations. The explicit form is derived of the equations that are satisfied by material and shape derivatives. The existence of weak material derivative is proved. These general findings are demonstrated on the steepest descent optimization procedure. Simulations involving the level set method for tracing the interface are performed for several materials with nonlinear heat conductivity. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical approximation of one phase quadrature domains

In this work, we present two numerical schemes for a free boundary problem that is called one phase quadrature domain. In the first method, using the properties of a given free boundary problem, we derive a method that leads us to a fast iterative solver. The iteration procedure is adapted to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient shape Quasi‐Newton method. Various numerical experiments confirm the efficiency of the derived numerical methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Risk averse elastic shape optimization with parametrized fine scale geometry

Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.     

Temperature dependence of an elastic modulus in generalized linear micropolar thermoelasticity

The model of the equations of generalized linear micropolar thermoelasticity with two relaxation times in an isotropic medium with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of reference temperature. Laplace and exponential Fourier transform techniques are used to obtain the solution by a direct approach. The integral transforms have been inverted by using a numerical technique to obtain the temperature, displacement, force and couple stress in the physical domain. The results of these quantities are given and illustrated graphically. A comparison is made with results obtained in case of temperature-independent modulus of elasticity. The problem of generalized thermoelasticity has been reduced as a special case of our problem.     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.