Ill-posed Boundary Value Problems Research Articles

Direct solution of ill-posed boundary value problems by radial basis function collocation method

Numerical solution of ill-posed boundary value problems normally requires iterative procedures. In a typical solution, the ill-posed problem is first converted to a well-posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill-posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.

Uniqueness studies in boundary value problems involving some second gradient models

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions.

Uniqueness studies in boundary value problems involving some second gradient models

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions.

A meso–macro finite element modelling of laminate structures: Part II: time-dependent behaviour

A meso–macro modelling is proposed for laminates made of unidirectional layers of a polymer matrix reinforced with long fibres. The time-independent behaviour introduced in the first part of this article is improved herein to account for viscous phenomena through viscoelasticity and viscoplasticity. A spectrum-type viscoelastic model is considered, which is based on the definition of elementary viscoelastic mechanisms. Its mathematical formulation is simplified by using a relaxation times triangular layout. A generalised Norton-type model integrating the elastic domain concept is used to report the plastic strains delay. A zero-valued “dynamic” yield function is incorporated into the traditional viscoplastic format, allowing a same treatment of plastic and viscoplastic problems. The integration of the layer behaviour through the thickness is obtained within a Kirchhoff shell finite element. The constitutive equations are integrated using two families of algorithms that generalise the well-known trapezoidal and mid-point rules, for which accuracy and non-linear stability analysis are carried out. Significant robustness of the local iterative solution is provided by complementing the basic Newton’s scheme with a local line-search strategy. In the case of a fully coupled plastic–viscoplastic behaviour, the local Newton’s iterative scheme is associated with a grid-search method in order to define available initial solutions. A perturbation technique is suggested to evaluate an algorithmic tangent operator since the viscoelasticity renders non-trivial an explicit determination of a consistent tangent operator. The proposed formulation has been implemented in the finite element code CASTEM2000 ® in order to test its validity. The obtained results are compared with semi-analytical ones in the case of progressive repeated loading tests by applying pure traction and pure pressure. Creep tests are also considered. To avoid ill-posed boundary value problems and to take account of a time-dependent damage process, viscous regularisation of the time-independent damage model is finally introduced with a structure analogous to the Perzyna-type viscoplasticity.

Regularized boundary element solution for an inverse boundary value problem in linear elasticity

AbstractIn this paper the Tikhonov regularization method is analysed for the ill‐conditioned system of linear equations which arises from the discretization of an ill‐posed boundary value problem in linear elasticity. The choice of the regularization parameter is achieved according to the L‐curve method. Then the regularization method is numerically implemented using the boundary element method (BEM). Copyright © 2002 John Wiley & Sons, Ltd.

An inversion method for parabolic equations based on quasireversibility

This paper is concerned with a new method to solve a linearized inverse problem for one-dimensional parabolic equations. The inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary. The method considers a temporal interval during which time dependent measurements are provided. It linearizes the working equation around the system response for a background medium. It is then possible to relate the inverse problem of interest to an ill-posed boundary value problem for a differential-integral equation, whose solution is obtained by the method of quasireversibility. This approach leads to an iterative method. A number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only.

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

Elastostatic inverse formulation

In this paper an inverse method for solving elastostatic problems with incomplete boundary conditions is presented. In general, inverse problems are ill-posed boundary value problems whose stability and uniqueness of solution and sensitivity-based formulations require additional constraints. In the development we use the Betti-reciprocal theorem to represent the boundary traction field in terms of the boundary and field displacements in an integral form. Initially, we assume the unknown boundary conditions and deformations required to solve the problem. In this way we equate the work done by the exact solution (unknown) to the work done by an assumed solution. Discretizing the resulting equations and using an iterative procedure each step in the solution process becomes the solution to a well-posed problem. Thus, with sufficient perturbations the correct boundary conditions are reconstructed.

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative 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. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.

On improperly posed Caucfay problems and their approximate solution

It is well known that, in general, the Cauchy problem for the Laplace equation does not allow a solution and therefore is ill-posed in both the Hadamard and the Tikhonov senses. The present work focuses on the question whether the problem has any meaningful approximate solution for arbitrary boundary conditions. Firstly, it is shown that it is possible to construct an analytic function which assumes some prescribed value on part of the boundary of a simply-connected domain. This problem is then shown to be equivalent to the Cauchy problem under consideration, the solution to which can thus be invariably approximated to any degree of accuracy on the unit circle centred at the origin when both the potential and the flux are specified as square-integrable functions over half the unit circle boundary. The uniqueness of the exact solution to the problem is also established. These results are actually true for any simply-connected domain which can be conformally mapped onto the unit circle so that the part of its boundary with prescribed potential and flux corresponds to one-half of the unit circle boundary. Finally, the feasibility of a boundary element formulation for a generic type of ill-posed boundary value problems is briefly discussed.

A study of the nonoscillatory character of solutions of partial differential equations by the method of separation of variables

The oscillatory character of solutions of partial differential equations is studied by the method of separation of variables; an ill-posed boundary value problem for a fourth-order polyharmonic equation, where the boundary conditions are given on two rectangles embedded one in the other, is studied; bounds for conditional stability and regularization are established.

A finite-element method for inertial waves in a frustum

A finite-element method has been developed to model inertial waves in a frustum of a cone, since analytical methods have proved inadequate. The governing Poincaré problem is posed as a variational principle and approximate eigensolutions are computed. The numerical results are used to complete the interpretation of the experimental results of Beardsley (1970) for a frustum of a cone. The significant role played by characteristic surfaces partly explains the enigmatic nature of the ill-posed boundary-value problem that describes inertial wave resonances.

Correct solvability of boundary-value problems in a halfspace for quasielliptic equations

B. I. Ptashnik, Ill-Posed Boundary Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984). Yu. N. Valitskii, "A four-point problem for a differential equation in a Banach space," Funkts. Anal. Prilozhen., 15, No. 4, 69-70 (1981). Yu. N. Valitskii, "The conditional well-posedness of a four-point problem for a certain differential equation," in: Questions of Well-Posedness for Inverse Problems of Mathematical Physics [in Russian], Vychisl. Tsentr. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk (1982), pp. 45-49. Yu. N. Valitskii, "On the well-posedness of a multipoint problem for a differential equation with operator coefficients," Dokl. Akad. Nauk SSSR, 286, No. 5, 1041-1043 ( 1 9 8 6 ) .

Regularization of a boundary value problem for an oscillation equation

An ill-posed boundary value problem for an oscillation equation is considered. The nature of this problem is discussed, a regularizing algorithm for a solution is constructed, and its interpretation in terms of cosine and sine-operator-functions is given.