A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations and involving a one-dimensional cooling element represented by a contour on which interface boundary conditions are specified. The problem consists in finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locally optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shape-differential calculus. The main novelty of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundary-integral formulation which exploits certain analytical properties of the solution and does not require grids adapted to the contour. This approach is thoroughly validated and optimization results obtained in different test problems exhibit nontrivial shapes of the computed optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing" (31 pages, 9 figures

An improved unified solver for compressible and incompressible fluids involving free surfaces. II. Multi-time-step integration and applications

An improved numerical solver for the unified solution of compressible and incompressible fluids involving interfaces is proposed. The present method is based on the CIP-CUP (Cubic Interpolated Propagation / Combined, Unified Procedure) method, which is a pressure-based semi-implicit solver for the Euler equations of fluid flows. In Part I of this series of articles [M. Ida, Comput. Phys. Commun. 132 (2000) 44], we proposed an improved scheme for the convection terms in the equations, which allowed us discontinuous descriptions of the density interface by replacing the cubic interpolation function used in the CIP scheme with a quadratic extrapolation function only around the interface. In this paper, as Part II of this series, the multi-time-step integration technique is adapted to the CIP-CUP integration. Because the CIP-CUP treats different-nature components in the fluid equations separately, the adaptation of the technique is straightforward. This modification allows us flexible determinations of the time interval, which results in an efficient and accurate integration. Furthermore, some additional discussion on our methods is presented. Finally, the application results to composite flow problems such as compressible and incompressible Kelvin-Helmholtz instabilities and the dynamics of two acoustically coupled deformable bubbles in a viscous liquid are provided.Comment: 34 pages, 13 figures, elsart; Typo in Eq.25 corrected; Publishe

A convex analysis approach to optimal controls with switching structure for partial differential equations

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

A Mixed PDE/Monte Carlo approach as an efficient way to price under high-dimensional systems

We propose to price derivatives modelled by multi-dimensional systems of stochastic di�fferential\ud equations using a mixed PDE/Monte Carlo approach. We derive a stochastic PDE where some of the coeffi�cients are conditional on stochastic ancillary factors. The stochastic\ud PDE is solved with either analytical or �finite diff�erence methods, where we simulate all the ancillary processes using Monte Carlo. The multilevel technique has also been introduced to further reduce the variance. The combined method showed over 80% cost reduction for the same accuracy, in pricing a barrier option in an FX market with stochastic interest rate and volatility (which is usually expensive to work with) , when compared to the pure Monte\ud Carlo simulation

Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations

We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.Comment: in Inverse Problems and Imaging Volume: 9, Number: 1 February 2015. arXiv admin note: text overlap with arXiv:1510.0752