Multidisciplinary Inverse Problems

This paper presents a limited survey of methods and multidisciplinary applications of various techniques for the solution of several classes of inverse problems as developed and practiced by our research team. Sketches of solution methods for inverse problems of shape determination, boundary conditions determination, sources determination, and physical properties determination are presented from the fields of aerodynamics, heat transfer, elasticity, and electrostatics. GENERAL INTRODUCTION Engineering field problems are defined (Kubo, 1993) by the governing partial differential or integral equation(s), shape(s) and size(s) of the domain(s), boundary and initial conditions, material properties of the media contained in the field, and by internal sources and external forces or inputs. If all of this information is known, the field problem is of an analysis (or direct) type and generally considered as well posed and solvable. If any of this information is unknown or unavailable, the field problem becomes an indirect (or inverse) problem and is generally considered to be ill posed and unsolvable. Specifically, inverse problems can be classified as: 1. Shape determination inverse problems, 2. Boundary/initial value determination inverse problems, 3. Sources and forces determination inverse problems, 4. Material properties determination inverse problems, and 5. Governing equation(s) determination inverse problems. The inverse problems are solvable if additional information is provided and if appropriate numerical algorithms are used. The objective of this paper is to offer a very brief survey of research on the solution methods for multidisciplinary inverse problems that has been performed in our Multidisciplinary Analysis, Inverse Design and Optimization (MAIDO) Laboratory. + Associate Professor. Fellow ASME. 1. SHAPE DETERMINATION INVERSE PROBLEMS The problem of determining sizes, shapes, and locations of objects or cavities inside a given object sounds like a formidable task. In reality, this type of inverse problem is probably the most common. The problem can be solved only if certain field quantity (pressure, heat flux, stress, magnetic field, etc.) can be specified on these unknown boundaries in addition to their complementary field quantities (velocity, temperature, deformation, electric field, etc.) on the same boundaries. 1.1 Aerodynamic Shape Inverse Design A typical inverse aerodynamic shape design is defined as follows: if a desirable fluid pressure distribution is specified on the yet unknown surface of an aerodynamic body, find the shape of the body that will produce this pressure distribution subject to the specified global flow-field conditions. Two classes of tools for inverse aerodynamic shape design are: a) methods with coupled shape modification and flow-field analysis, and b) methods with uncoupled shape modification and flow-field analysis (Dulikravich, 1984; 1987; 1991; 1992; 1995; 1997; Fujii and Dulikravich, 1999; Tanaka and Dulikravich, 1998). The coupled inverse shape design methods require special consideration in the writing of the flow-field analysis computer code. These software modifications represent a major undertaking, even if the source version of the flow-field analysis code is available. For example, an indirect surface transpiration technique might need to exchange no-slip wall boundary conditions on the body surface with specified pressure boundary conditions in order to obtain a shape update. Other examples of this class of design techniques are: stream-function-ascoordinate formulation, characteristic boundary condition concept, integro-differential equation concept, fictitious gas concept, direct surface transpiration concept, and adjoint operator/control theory approaches. Furthermore, most of the existing inverse shape design methods are not applicable to viscous flows or to three-dimensional configurations.