Simultaneous numerical determination of a corroded boundary and its admittance

In this paper, an inverse geometric problem for Laplace’s equation arising in boundary corrosion detection is considered. This problem, which consists of determining an unknown corroded portion of the boundary of a bounded domain and its admittance Robin coefficient from two pairs of boundary Cauchy data (boundary temperature and heat flux), is solved numerically using the meshless method of fundamental solutions. A non-linear minimization of the objective function is regularized, and the stability of the numerical results is investigated with respect to noise in the input data and various values of the regularization parameters involved

The Reciprocity Gap Functional for Identifying Defects and Cracks

International audienceThe recovery of defects and cracks in solids using overdetermined boundary data, both the Dirichlet and the Neumann types, is considered in this paper. A review of the method for solving these inverse problems is given, focusing particularly on linearized inverse problems. It is shown how the reciprocity gap functional can solve nonlinear inverse problems involving identification of cracks and distributed defects in bounded solids. Exact solutions for planar cracks in 3D solids are given for static elasticity, heat diffusion and transient acoustics

Study on steady-state thermal conduction with singularities in multi-material composites

Increasing demand in material and mechanical properties has led to production of complex composite structures. The composite structures, made of different materials, possess a variety of properties derived from each material. This has brought challenges in both analytical and numerical studies in thermal conduction which is of significant importance for thermoelastic problems. Therefore, a unified and effective approach would be desirable. The present study makes a first attempt to determining the analytical symplectic eigen solution for steady-state thermal conduction problem of multi-material crack. Based on the obtained symplectic eigen solution (including higher order expanding eigen solution terms), a new symplectic analytical singular element (SASE) for numerical modeling is constructed. It is concluded that composite structures composed of multi-material with complex geometric shapes can be modeled by the developed method, and the generalized flux intensity factors (GFIFs) can be solved accurately and efficiently

Solution of inverse problem - regularization via thermodynamical criterion

In engineering practice, measuring temperature on both sides of a wall (of, for example, turbine casing or combustion chamber) is not always possible. On the other hand, measurement of both temperature and heat flux on the outer surface of the wall is possible. For transient heat conduction equation, measurements of temperature and heat flux supplemented by the initial condition state the Cauchy problem, which is ill-conditioned In this paper, the stable solution is obtained for the Cauchy problem using the Laplace transformation and the minimisation of continuity in the process of integration of convolution. Test examples confirm proposed algorithm for the inverse problem solution.Papers presented to the 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Costa de Sol, Spain on 11-13 July 2016

Novel optimisation methods for numerical inverse problems

Inverse problems involve the determination of one or more unknown quantities usually appearing in the mathematical formulation of a physical problem. These unknown quantities may be boundary heat flux, various source terms, thermal and material properties, boundary shape and size. Solving inverse problems requires additional information through in-situ data measurements of the field variables of the physical problems. These problems are also ill-posed because the solution itself is sensitive to random errors in the measured input data. Regularisation techniques are usually used in order to deal with the instability of the solution. In the past decades, many methods based on the nonlinear least squares model, both deterministic (CGM) and stochastic (GA, PSO), have been investigated for numerical inverse problems. The goal of this thesis is to examine and explore new techniques for numerical inverse problems. The background theory of population-based heuristic algorithm known as quantum-behaved particle swarm optimisation (QPSO) is re-visited and examined. To enhance the global search ability of QPSO for complex multi-modal problems, several modifications to QPSO are proposed. These include perturbation operation, Gaussian mutation and ring topology model. Several parameter selection methods for these algorithms are proposed. Benchmark functions were used to test the performance of the modified algorithms. To address the high computational cost of complex engineering optimisation problems, two parallel models of the QPSO (master-slave, static subpopulation) were developed for different distributed systems. A hybrid method, which makes use of deterministic (CGM) and stochastic (QPSO) methods, is proposed to improve the estimated solution and the performance of the algorithms for solving the inverse problems. Finally, the proposed methods are used to solve typical problems as appeared in many research papers. The numerical results demonstrate the feasibility and efficiency of QPSO and the global search ability and stability of the modified versions of QPSO. Two novel methods of providing initial guess to CGM with approximated data from QPSO are also proposed for use in the hybrid method and were applied to estimate heat fluxes and boundary shapes. The simultaneous estimation of temperature dependent thermal conductivity and heat capacity was addressed by using QPSO with Gaussian mutation. This combination provides a stable algorithm even with noisy measurements. Comparison of the performance between different methods for solving inverse problems is also presented in this thesis

Singular Superposition/Boundary Element Method for Reconstruction of Multi-dimensional Heat Flux Distributions with Application to Film Cooling Holes

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given initial strength distribution. A forward steady-state heat conduction problem is solved using the boundary element method (BEM), and an objective function is defined to measure the difference between the heat flux measured at the exposed surfaces and the heat flux predicted by the BEM under the current strength distribution of the singularities. A Genetic Algorithm (GA) iteratively alters the strength distribution of the singularities until the measuring surfaces heat fluxes are matched, thus satisfying Cauchy conditions. The distribution of the heat flux at the walls of the cooling hole is determined in a post-processing stage after the inverse problem is solved. The advantage of this technique is to eliminate the need of meshing the surfaces of the cooling holes, which requires a large amount of effort to achieve a high quality mesh. Moreover, the use of singularity distributions significantly reduces the number of parameters sought in the inverse problem, which constitutes a tremendous advantage in solving the inverse problem, particularly in the application of film cooling holes

Numerical solution of inverse problems in mechanics using the boundary element method

Due to the ill-posed nature of inverse problems, it is difficult to obtain solutions using well known analytical and numerical techniques. The use of boundary element method as a numerical technique to solve inverse problems is quite new. In this work, the algorithms for the solution of two kinds of inverse problems are examined in detail. For the first kind, the shape and location of a part of the boundary is unknown; and for the second kind, the boundary condition is not specified on a part of the boundary;Boundary value problems with partially unknown boundary are ill-posed. To solve these problems additional information is necessary. Over-specified boundary data in the form of experimentally measured quantities can be used as additional information for solving the problem. An algorithm, based on the boundary element method and non-linear optimization techniques, is proposed to solve this inverse problem. Using the overspecified boundary data, a functional is formed which involves parameters describing the unknown boundary. Minimization of this functional with respect to these parameters determines the unknown boundary. The performance of this scheme is examined through two problems. It is shown that the algorithm performs well even for complex shapes of the unknown boundary;For the problems in which the specified boundary conditions are insufficient, experimentally obtained data at some internal points are used as additional conditions. The boundary is divided into straight boundary elements and the unknown boundary conditions are represented as unknowns at the nodes of the boundary elements. It is shown that, for practical reasons, the number of nodes where the boundary condition is not specified is usually larger than the number of probes used for obtaining interior data. This results in an under-determined system of linear equations. A regularization method is used to solve these equations. The scheme, when applied to several example problems, showed satisfactory performance. Few guidelines for the placement of the temperature probes in the interior of the domain are developed through numerical experiments

Self-consistent stationary MHD shear flows in the solar atmosphere as electric field generators

Magnetic fields and flows in coronal structures, for example, in gradual phases in flares, can be described by 2D and 3D magnetohydrostatic (MHS) and steady magnetohydrodynamic (MHD) equilibria. Within a physically simplified, but exact mathematical model, we study the electric currents and corresponding electric fields generated by shear flows. Starting from exact and analytically calculated magnetic potential fields, we solveid the nonlinear MHD equations self-consistently. By applying a magnetic shear flow and assuming a nonideal MHD environment, we calculated an electric field via Faraday's law. The formal solution for the electromagnetic field allowed us to compute an expression of an effective resistivity similar to the collisionless Speiser resistivity. We find that the electric field can be highly spatially structured, or in other words, filamented. The electric field component parallel to the magnetic field is the dominant component and is high where the resistivity has a maximum. The electric field is a potential field, therefore, the highest energy gain of the particles can be directly derived from the corresponding voltage. In our example of a coronal post-flare scenario we obtain electron energies of tens of keV, which are on the same order of magnitude as found observationally. This energy serves as a source for heating and acceleration of particles.Comment: 11 pages, 9 figures, accepted to Astronomy and Astrophysic