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

Inverse Heat Conduction Methods in the CHAR Code for Aerothermal Flight Data Reconstruction

Reconstruction of flight aerothermal environments often requires the solution of an inverse heat transfer problem, which is an ill-posed problem of determining boundary conditions from discrete measurements in the interior of the domain. This paper will present the algorithms implemented in the CHAR code for use in reconstruction of EFT-1 flight data and future testing activities. Implementation details will be discussed, and alternative hybrid-methods that are permitted by the implementation will be described. Results will be presented for a number of problems

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

Challenges for the Accurate Determination of the Surface Thermal Condition via In-Depth Sensor Data

The overall goal of this work is to provide a systematic methodology by which the difficulties associated with the inverse heat conduction problem (IHCP) can be resolved. To this end, two inverse heat conduction methods are presented. First, a space-marching IHCP method (discrete space, discrete time) utilizing a Gaussian low-pass filter for regularization is studied. The stability and accuracy of this inverse prediction is demonstrated to be more sensitive to the temporal mesh than the spatial mesh. The second inverse heat conduction method presented aims to eliminate this feature by employing a global time, discrete space inverse solution methodology. The novel treatment of the temporal derivative in the heat equation, combined with the global time Gaussian low-pass filter provides the regularization required for stable, accurate results. A physical experiment used as a test bed for validation of the numerical methods described herein is also presented. The physics of installed thermocouple sensors are outlined, and loop-current step response (LCSR) is employed to measure and correct for the delay and attenuation characteristics of the sensors. A new technique for the analysis of LCSR data is presented, and excellent agreement is observed between this model and the data. The space-marching method, global time method, and a new calibration integral method are employed to analyze the experimental data. First, data from only one probe is used which limits the results to the case of a semi-infinite medium. Next, data from two probes at different depths are used in the inverse analysis which enables generalization of the results to domains of finite width. For both one- and two-probe analyses, excellent agreement is found between the actual surface heat flux and the inverse predictions. The most accurate inverse technique is shown to be the calibration integral method, which is presently restricted to one-probe analysis. It is postulated that the accuracy of the global time method could be improved if the required higher-time derivatives of temperature data could be more accurately measured. Some preliminary work in obtaining these higher-time derivatives of temperature from a voltage-rate interface used in conjunction with the thermocouple calibration curve is also presented

Quantitative non-destructive testing

The work undertaken during this period included two primary efforts. The first is a continuation of theoretical development from the previous year of models and data analyses for NDE using the Optical Thermal Infra-Red Measurement System (OPTITHIRMS) system, which involves heat injection with a laser and observation of the resulting thermal pattern with an infrared imaging system. The second is an investigation into the use of the thermoelastic effect as an effective tool for NDE. As in the past, the effort is aimed towards NDE techniques applicable to composite materials in structural applications. The theoretical development described produced several models of temperature patterns over several geometries and material types. Agreement between model data and temperature observations was obtained. A model study with one of these models investigated some fundamental difficulties with the proposed method (the primitive equation method) for obtaining diffusivity values in plates of thickness and supplied guidelines for avoiding these difficulties. A wide range of computing speeds was found among the various models, with a one-dimensional model based on Laplace's integral solution being both very fast and very accurate

Convection and AGN Feedback in Clusters of Galaxies

A number of studies have shown that the convective stability criterion for the intracluster medium (ICM) is very different from the Schwarzchild criterion due to the effects of anisotropic thermal conduction and cosmic rays. Building on these studies, we develop a model of the ICM in which a central active galactic nucleus (AGN) accretes hot intracluster plasma at the Bondi rate and produces cosmic rays that cause the ICM to become convectively unstable. The resulting convection heats the intracluster plasma and regulates its temperature and density profiles. By adjusting a single parameter in the model (the size of the cosmic-ray acceleration region), we are able to achieve a good match to the observed density and temperature profiles in a sample of eight clusters. Our results suggest that convection is an important process in cluster cores. An interesting feature of our solutions is that the cooling rate is more sharply peaked about the cluster center than is the convective heating rate. As a result, in several of the clusters in our sample, a compact cooling flow arises in the central region with a size R that is typically a few kpc. The cooling flow matches onto a Bondi flow at smaller radii. The mass accretion rate in the Bondi flow is equal to, and controlled by, the rate at which mass flows in through the cooling flow. Our solutions suggest that the AGN regulates the mass accretion rate in these clusters by controlling R: if the AGN power rises above the equilibrium level, R decreases, the mass accretion rate drops, and the AGN power drops back down to the equilibrium level.Comment: 41 pages, 7 figures, accepted for publication in ApJ. Changes in this version: extended discussion of Bondi accretion in clusters, better mass model, new numerical solution

Microscale rarefied gas dynamics and surface interactions for EUVL and MEMS applications.

A combined experimental/modeling study was conducted to better understand the critical role of gas-surface interactions in rarefied gas flows. An experimental chamber and supporting diagnostics were designed and assembled to allow simultaneous measurements of gas heat flux and inter-plate gas density profiles in an axisymmetric, parallel-plate geometry. Measurements of gas density profiles and heat flux are made under identical conditions, eliminating an important limitation of earlier studies. The use of in situ, electron-beam fluorescence is demonstrated as a means to measure gas density profiles although additional work is required to improve the accuracy of this technique. Heat flux is inferred from temperature-drop measurements using precision thermistors. The system can be operated with a variety of gases (monatomic, diatomic, polyatomic, mixtures) and carefully controlled, well-characterized surfaces of different types (metals, ceramics) and conditions (smooth, rough). The measurements reported here are for 304 stainless steel plates with a standard machined surface coupled with argon, helium, and nitrogen. The resulting heat-flux and gas-density-profile data are analyzed using analytic and computational models to show that a simple Maxwell gas-surface interaction model is adequate to represent all of the observations. Based on this analysis, thermal accommodation coefficients for 304 stainless steel coupled with argon, nitrogen, and helium are determined to be 0.88, 0.80, and 0.38, respectively, with an estimated uncertainty of {+-}0.02