Inverse methods (original) (raw)

An inverse method for finding unknown surface tractions and deformations in elastostatics

Computers & Structures, 1995

We have developed a non-iterative algorithm for determining unknown deformations and tractions on surfaces of arbitrarily shaped solids where these quantities cannot be measured or evaluated. For this inverse boundary value technique to work, both deformations and tractions must be available and applied simultaneously on at least a part of the object's surface called an over-specified boundary. Our method is non-iterative only because it utilizes the boundary element method (BEM) to calculate deformations and tractions on surfaces where they are unavailable and simultaneously computes the stress and deformation field within the entire object. Inversely computed displacement and stress fields within simple solids and on their boundaries were in excellent agreement with the BEM analysis results and analytic solutions. Our algorithm is highly flexible in treating complex geometries and mixed elastostatics boundary conditions. The accuracy and reliability of this technique deteriorates when the known surface conditions are only slightly over-specified and far from the inaccessible surfaces.

Numerical solutions in three dimensional elastostatics

International Journal of Solids and Structures, 1969

A numerical solution capability is developed for the solution of problems in three dimensional elastostatics. The solution method utilizes singular integral equations which can be solved numerically for the unknown surface tractions and displacements for the fully mixed boundary value problem. The method is independent of the surface shape and data specification and has been fully automated. Some sample problems are solved to verify the formulation. In addition the method has been used to investigate a significant problem with stress singularities.

Efficient Iterative Solution for Large Elasto-Dynamic Inverse Problems

JSME International Journal Series A, 2004

Convergence of the solver and the regularization are two important issues concerning an ill-posed inverse problem. The intrinsic regularization of the conjugate gradient method along iteration makes the method superior for solving an ill-posed problem. The solutions along iteration converge fast to an optimal solution. If the termination criterion is not satisfied, the solution will diverge to a solution which dominated by the noise. Reformulation of an ill-posed problem as an eigenvalue formulation gives a very convenient formula since it is possible to estimate an optimal regularization parameter and an optimal solution at once. For very large problems, the fast Fourier transformation could be implemented in the circulant matrix-vector multiplication. The developed method is applied to some inverse problems of elasto-dynamic and the accurate estimation was achieved.

Inverse Determination of Steady Boundary Conditions in Heat Transfer and Elasticity

In the case of steady heat conduction and steady elasticity there is often a problem of not knowing temperatures and heat fluxes or tractions and deformations boundary conditions on some parts of the boundary. However, there should be a sufficient amount of over-specified boundary conditions available on at least a portion of the remaining boundary. . In this case, the entire process of determining the boundary conditions is of a truly non-destructive nature since there is no need to have values of the field variables at points inside the object. These problems can be solved either non-iteratively using a boundary element formulation or iteratively using a finite element formulation. In the case of a non-iterative formulation the solution process is very simple and numerical results are practically guaranteed. In the case of an iterative formulation, a judicious application of appropriate regularization algorithms must be performed. Furthermore, the over-specified boundary conditions must be highly accurate. Both algorithms have been demonstrated to work on simply-connected and multiply-connected two-and-three-dimensional configurations.

An inverse problem of elastostatics in mechanics of composites

Composites Science and Technology, 2008

In this study we consider an inverse problem of elastostatics and its applications appearing in mechanics of composites. In many cases surface displacements can be monitored on a part of a stress-free boundary of an elastic composite (in general, heterogeneous). When this information is further used for stress analysis it leads to redundancy in boundary conditions on the part where displacements have been measured. To compensate this redundancy no boundary conditions are imposed on some internal boundaries such as cracks, inclusions or interfaces between dissimilar materials in a particular composite. As the result one arrives to an ill-posed boundary value problem of elasticity overspecified on a part of the entire boundary and underspecified on the rest of it. This paper presents general approach based on integral equations and studies one particular example for the reconstruction of characteristics of narrow process zones developing near the crack tips.

Quantitative elastography, solving the inverse elasticity problem using the Gauss-Newton method

2008 IEEE Ultrasonics Symposium, 2008

This work presents the application of a constrained Gauss Newton method for the solution of an inverse elasticity problem in ultrasound elastography. This algorithm was written taking care of real constraints, like the limited and noisy data used to estimate the volume's properties and the speed necessary to have a real time application in a real medical environment. The algorithm is tested on data acquired on polyvinyl alcohol (PVA) phantoms. The role of the amount of elements used to discretized the volume, the role of the regularization and the role of noise are investigated. The possible application of the algorithm is in the use of quantitative elastography as tactile sensor for tactile feedback in minimally invasive surgery.

An inverse method for determining elastic material properties and a material interface

International Journal for Numerical Methods in Engineering, 1992

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.

Error analysis of the method of fundamental solutions for linear elastostatics

Journal of Computational and Applied Mathematics, 2013

For linear elastostatics in 2D, the Trefftz methods (i.e., the boundary methods) using the particular solutions and the fundamental solutions satisfying the Cauchy-Navier equation lead to the method of particular solutions (MPS) and the method of fundamental solutions (MFS), respectively. In this paper, the mixed types of the displacement and the traction boundary conditions are dealt with, and both the direct collocation techniques and the Lagrange multiplier are used to couple the boundary conditions. The former is just the MFS and the MPS, and the latter is also called the hybrid Trefftz method (HTM) in Jirousek (1978, 1992, 1996) [1-3]. In Bogomolny (1985) [4] and Li (2009) [5] the error analysis of the MFS is given for Laplace's equation, and in Li (2012) [6] the error bounds of both MPS and HTM using particular solutions (PS) are provided for linear elastostatics. In this paper, our efforts are devoted to explore the error analysis of the MFS and the HTM using fundamental solutions (FS). The key analysis is to derive the errors between FS and PS of the linear elastostatics, where the expansions of the FS in Li et al. (2011) [7] are a basic tool in analysis. Then the optimal convergence rates can be achieved for the MFS and the HTM using FS. Recently, the MFS has been developed with numerous reports in computation; the analysis is behind. The analysis of the MFS for linear elastostatics in this paper may narrow the existing gap between computation and theory of the MFS.