An iterative boundary element method for Cauchy inverse problems

 We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. A new inversion method is introduced. It reduces the Cauchy problem resolution to the determination of the resolution of a sequence of well-posed problems. The sequence of these solutions is proved to converge to the Cauchy problem solution. The algorithm is implemented in the framework of boundary elements. Displayed numerical results highlight its accuracy, as well as its robustness to noisy data. Received 6 November 2000     

A boundary elements pseudo heat source method formulation for inverse analysis of solidification problems

 An efficient methodology is presented to solve inverse solidification problems. In the procedure, the latent heat effects are implemented by introducing pseudo heat sources near the moving interface. The material properties can be temperature dependent. To account for the nonlinear part of the governing differential equations, a finite-boundary element formulation is employed. To reduce the oscillations in the solution, a sequential regularization scheme is used. A procedure for proper selection of regularization parameters is presented. To smooth the solutions further, a secondary regularization scheme is introduced and employed. Two complete examples are presented to demonstrate the applicability and the accuracy of the methods. Received: 1 March 2002 / Accepted: 10 February 2003     

Evolutionary optimization in thermoelastic problems using the boundary element method

 The paper is devoted to application of evolutionary algorithms and the boundary element method to shape optimization of structures for various thermomechanical criteria, inverse problems of finding an optimal distribution of temperature on the boundary and identification of unknown boundary. Design variables are specified by Bezier curves. Several numerical examples of evolutionary computation are presented. Received 6 November 2000     

The ponding problem on elastic membranes: an analog equation solution

 In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 28 July 2001     

A well-conditioned technique for solving the inverse problem of boundary traction estimation for a constrained vibrating structure

 Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L₂-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement. Received: 10 January 2002 / Accepted: 24 October 2002 This work was performed under contract from the Swedish Defence Material Administration (FMV). The funding provided is gratefully acknowledged. The author would like to thank all staff at the Structural Dynamics research group, Department of Structures and Materials, Aeronautics Division, Swedish Defence Research Agency.     

Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

This paper describes some recent observations associated with (1) clarifying and expanding upon the integral relationship between temperature and heat flux in a half-space; (2) offering an analytic-continuation approach for estimating the surface temperature and heat flux in a one-dimensional geometry based on embedded measurements; and, (3) offering a novel digital filter that supports the use of analytic continuation based on a minimal number of embedded sensors. Key to future inverse analysis must be the proper understanding and generation of rate data associated with both the temperature and heat flux at the embedded location. For this paper, some results are presented that are theoretrically motivated but presently adapted to implement digital filtering. A pulsed surface heat flux is reconstructed by way of a single thermocouple sensor located at a well-defined embedded location in a half space. The proposed low-pass, Gaussian digital filter requires the specification of a cut-off frequency that is obtained by viewing the power spectra of the temperature signal as generated by the Discrete Fourier Transform (DFT). With this in hand, and through the use of an integral relationship between the local temperature and heat flux at the embedded location, the embedded heat flux can be accurately estimated. The time derivatives of the filtered temperature and heat flux are approximated by a simple finite-difference method to provide a sufficient number of terms required by the Taylor series for estimating (i.e., the projection) the surface temperature and heat flux. A numerical example demonstrates the accuracy of the proposed scheme. A series of appendices are offered that describe the mathematical details omitted in the body for ease of reading. These appendices contain important and subtle details germane to future studies.     

Solution of a multidimensional inverse radiation problem by means of mode reduction

The Karhunen–Loève Galerkin procedure is employed to solve an inverse radiation problem of determining the time‐varying strength of a heat source, which mimics flames in a furnace, from temperature measurements in three‐dimensional participating media where radiation and conduction occur simultaneously. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, using a conjugate gradient method. Through the Karhunen–Loève Galerkin procedure, one can represent the system dynamics with a minimum degree of freedom, and consequently the amount of computation required in the solution of the inverse problem is reduced drastically when the present technique is adopted. Copyright © 2004 John Wiley & Sons, Ltd.     

Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations

 In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem for Helmholtz-type equations is analysed. An efficient regularising stopping criterion for CGM proposed by Nemirovskii [25] is employed. The numerical results obtained confirm that the CGM + BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. Received: 5 November 2002 / Accepted: 5 March 2003 L. Marin would like to acknowledge the financial support received from the EPSRC. The authors would like to thank Professor Dinh Nho Hào and Dr. Thomas Johansson for some useful discussions and suggestions.     

A statistical average algorithm for the dynamic compound inverse problem

 A new method named as Statistical Average Algorithm is developed for solving the structural dynamic compound inverse problem, which means to identify structural parameters with unknown input or to inverse input time history with unknown structural parameters. By taking the mechanical characteristic of the ground motion as an additional condition, an iterative algorithm based on the least-squares technique is developed so that the input process and structural parameters can be correctly determined using only output measurements. The procedure of the proposed method is discussed in detail, and a mathematical proof is presented as well in the appendix. A practical input situation such as earthquake, ambient vibration is considered in the numerical examples to verify the accuracy, reliability and robustness of the proposed algorithm. Considering the background of the practical application, both of the noise-free and noise-included output responses are considered in the numerical examples. In all cases, the proposed method identifies the structural parameters and reconstructs the input process rationally. Received: 12 March 2001 / Accepted: 16 September 2002     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli

 The radial basis functions (RBFs) have been proven to have excellent properties for interpolation problems, which can be considered as an efficient scheme for function approximation. In this paper, we will explore another type of approximation problem, that is, the derivative approximation, by the RBFs. A new approach, which is based on the differential quadrature (DQ) approximation for the derivative with RBFs as test functions, is proposed to approximate the first, second, and third order derivatives of a function. The performance of three commonly-used RBFs for some typical expressions of derivatives as well as the computation of one-dimensional Burgers equation are studied. Furthermore, the proposed method is applied to simulate natural convection in a concentric annulus by solving Navier–Stokes equations. The obtained results are compared well with exact data or benchmark solutions. Received: 27 June 2001 / Accepted: 29 July 2002     

Error estimates for isoparametric mixed finite element solution of 4th order elliptic problems with variable coefficients

 Considering the combined effect of boundary approximation and numerical integration, error estimates for the isoparametric mixed finite element solution of fourth order elliptic problems with variable coefficients in convex domains, which, in the particular case of aniso-/ortho-/ isotropic plate bending problems, gives a direct and simultaneous approximation to bending moment tensor field Ψ=(ψ _ij ) _1≤i,j≤2 and displacement field `u', have been developed. Results of numerical experiments justify the theoretical results. Received 20 July 2001 / Accepted 28 November 2001     

Real-time identification of a high-magnitude boundary heat flux on a plate

Identification of high-magnitude heat flux in real time is a challenging problem, since most of the currently available algorithms require large computation time in comparison with the time scale of the real physical problem. This paper presents a methodology that allows for quantifying the unsteady heat flux in real time by using the steady-state Kalman filter. Two different cases have been used to verify this algorithm and the estimates are in excellent agreement with the reference values.     

On using enriched cover function in the Partition-of-unity method for singular boundary-value problems

 Amongst the various approaches of `meshless' method, the Partition-of-unity concept married with the traditional finite-element method, namely PUFEM, has emerged to be competitive in solving the boundary-value problems. It inherits most of the advantages from both techniques except that the beauty of being `meshless' vanishes. This paper presents an alternative approach to solve singular boundary-value problems. It follows the basic PUFEM procedures. The salient feature is to enhance the quality of the influence functions, either over one single nodal cover or multi-nodal-covers. In the vicinity of the singularity, available asymptotic analytical solution is employed to enrich the influence function. The beauty of present approach is that it facilitates easy replacement of the influence functions. In other words, it favors the `influence-function refinement' procedure in a bid to search for more accurate solutions. It is analogous to the `p-version refinement' in the traditional finite-element procedures. The present approach can yield very accurate solution without adopting refined meshes. As a result, the quantities around the singularity can be evaluated directly once the nodal values are solved. No additional post-processing is needed. Firstly, the formulation of the present PUFEM approach is described. Subsequently, illustrative examples show the application to three classical singular benchmark problems having various orders of singularity. Results obtained through mesh refinements, single-nodal-cover refinements or multi-nodal-cover refinements are compared. Received: 5 November 2001 / Accepted: 21 May 2002     

Direct computation of instability points for contact problems

 The extended system is known as a reliable algorithm for the direct computation of instability points on the equilibrium path of mechanical structures. This article describes the application of the extended system as critical point computation method to mechanical contact problems. In this type of problems inequality constraints have to be considered. Moreover a prediction method based on the extended system algorithm is presented which allows the detection of favorable starting values for a critical point computation on the equilibrium path. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

Body-force-driven multiplicity and stability of combined free and forced convection in rotating curved ducts: Coriolis force

 A numerical study is made on the fully developed bifurcation structure and stability of forced convection in a rotating curved duct of square cross-section. Solution structure is determined as variation of a parameter that indicates the effect of rotation (Coriolis-force-driven multiplicity). Three solutions for the flows in a stationary curved duct obtained in the work of Yang and Wang [1] are used as initial solutions of continuation calculations to unfold the solution branches. Twenty-one solution branches are found comparing with five obtained by Selmi and Nandakumar [2]. Dynamic responses of the multiple solutions to finite random disturbances are examined by the direct transient computation. Results show that characteristics of physically realizable fully developed flows changes significantly with variation of effect of rotation. Fourteen sub-ranges are identified according to characteristics of physically realizable solutions. As rotation effect changes, possible physically realizable fully-developed flows can be stable steady 2-cell state, stable multi-cell state, temporal periodic oscillation between symmetric/asymmetric 2-cell/4-cell flows, temporal oscillation with intermittency, temporal chaotic oscillation and temporal oscillation with pseudo intermittency. Among these possible physically realizable fully developed flows, stable multi-cell state and stable steady 2-cell state exist as dual stable. And oscillation with pseudo intermittency is a new phenomenon. In addition to the temporal oscillation with intermittency, sudden shift from stationary stable solution to temporal chaotic oscillation is identified to be another way of onset of chaos. Received: 20 December 2001 / Accepted: 5 August 2002 The financial support from the Research Grants Council of Hong Kong (RGC, Project No: HKU7086/00 E), the CRCG and the Outstanding Young Researcher Award of the University of Hong Kong to LW is gratefully acknowledged.     

A semi-analytical approach to three-dimensional normal contact problems with friction

 We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions. Received: 10 July 2002 / Accepted: 3 December 2002 The authors appreciate the support of the UC Department of Mechanical Engineering and the UC Office of the Vice President for Research, who jointly provided funds for this work.     

Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods

 We look at the task of computing the time-evolution of a non-linear system for a long time, in our case under random external influences. Our specific example is the fatigue evaluation of a wind turbine. To facilitate such a computation, we look at a reduction of the computational effort by projecting everything on a low-dimensional basis. In this case we take the Karhunen-Loève basis generated from running the model a little while under the random loading. It is important that the error which is caused by this reduction process can be controlled. We estimate the error by dual or adjoint methods. This in turn allows the process of model reduction to be performed adaptively. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

Numerical integration in 3D Galerkin BEM solution of HBIEs

 We consider hypersingular boundary integral equations associated with 3D problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. At first, for linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations. Then, after an analysis of the singularities arising in the whole integration process, we propose suitable quadrature schemes to evaluate integrals required to form the Galerkin matrix elements. Several numerical results are presented. Received 6 November 2000     

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

 A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method. Received 16 December 2000 / Accepted 25 April 2002