An old elasticity problem in a unilateral setting

Soft tissue pathologies are often associated with changes in mechanical properties. For example, breast and other tumors usually present as stiff lumps. Imaging the spatial distribution of the mechanical properties of tissues thus reveals information of diagnostic value. Doing so, however, typically requires the solution of an inverse elasticity problem. In this work we consider the inverse elasticity problem for an incompressible material in plane stress, formulated and solved as a constrained optimization problem. We formulate this inverse problem enforcing high order continuity for our variables. Driven by the requirements for the strong and weak solutions to this problem, we assume that our data field (i.e. the measured displacement) is in H2 and our parameter distribution (i.e. the sought shear modulus distribution) is in H1. This high order regularity requirement for the data is incompatible with standard FEM. We solve this problem using a FEM formulation that is novel in two respects. First, we employ quadratic b-splines that enforce C1 continuity in our displacement field, consistent with the variational requirements of the continuous problem. Second, we include Galerkin-least-squares (GLS) stabilization in the iterative optimization formulation. GLS adds consistent stability to the discrete formulation that otherwise violates an ellipticity condition that is satisfied by the continuous problem. Computational examples validate this formulation and demonstrate numerical convergence with mesh refinement.

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Wavelet Solutions to the Natural Integral Equations of the Plane Elasticity Problem

Proceedings of the Second ISAAC Congress - International Society for Analysis, Applications and Computation ◽

10.1007/978-1-4613-0271-1_68 ◽

2000 ◽

pp. 1471-1480 ◽

Cited By ~ 2

Author(s):

W. Lin ◽

Y. J. Shen

Keyword(s):

Integral Equations ◽

Plane Elasticity ◽

Elasticity Problem ◽

Plane Elasticity Problem

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ROLE OF KUHN-TUCKER CONDITIONS IN ELASTICITY EQUATIONS IN TERMS OF STRESSES

Journal of Civil Engineering and Management ◽

10.3846/13921525.2000.10531573 ◽

2000 ◽

Vol 6 (2) ◽

pp. 104-112

Author(s):

Ela Chraptovič ◽

Juozas Atkočiūnas

Keyword(s):

Mathematical Programming ◽

System Integration ◽

Minimum Principle ◽

Functional Space ◽

Elasticity Problem ◽

Mathematical Programming Problem ◽

Variation Principle ◽

Three Dimension ◽

Strain Compatibility ◽

Elasticity Equations

Solution of the elasticity problem in terms of stresses leads to the stress vector six components, satisfying the Beltrami compatibility eqns and boundary conditions, evaluation. A direct integration of the nine differential eqns system in respect of the six stress components is difficult to realise practically. This is the reason why often the Casigliano variation principle to solve the boundary elasticity problem in terms of stresses is applied. An application of the above-mentioned principle ensures the satisfaction of all the six Saint-Venant strain compatibility eqns (see the works of Southwell, Kliushnikov, a.o.). Castigliano variation principle does not define the number of independent strain compatibility eqns. Thus, it is not clear whether the elasticity problem eqns system in terms of stresses is over-defined or not. The strain compatibility eqns for an ideal elastic body is investigated in the article by means of the mathematical programming theory. A mathematical model to evaluate the statically admissible stresses is formulated on the basis of complementary energy minimum principle. It is proved that the strain compatibility eqns mean the Kuhn-Tucker optimality conditions of the mathematical programming problem. The method to formulate the strain compatibility eqns in respect of the statically admissible stresses defining eqns formulation technique is revealed. The proposed method is illustrated to achieve the six component stresses vector in functional space for the three-dimension problem: usually the solution of the elasticity problem in terms of the stresses is realised via the nine eqns system integration. The Kuhn-Tucker conditions allowed to confirm an original but not usually applied Washizu conclusion about Cauchy geometrical compatibility eqns.

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Wavelet Algorithm for the Numerical Solution of Plane Elasticity Problem

Wavelet Analysis and Its Applications - Lecture Notes in Computer Science ◽

10.1007/3-540-45333-4_18 ◽

2001 ◽

pp. 139-144 ◽

Cited By ~ 1

Author(s):

Youjian Shen ◽

Wei Lin

Keyword(s):

Numerical Solution ◽

Plane Elasticity ◽

Elasticity Problem ◽

Plane Elasticity Problem ◽

Wavelet Algorithm

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Solving Topology Optimization Problems Using Wavelet Approximations

Volume 2: 24th Design Automation Conference ◽

10.1115/detc98/dac-5618 ◽

1998 ◽

Author(s):

Giuseppe C. A. DeRose ◽

Alejandro R. Díaz

Keyword(s):

Finite Element ◽

Topology Optimization ◽

Finite Element Methods ◽

Condition Number ◽

Optimization Problems ◽

Mesh Size ◽

Elasticity Problem ◽

New Method ◽

Galerkin Scheme

Abstract A new method to solve topology optimization problems is discussed. This method is based on the use of a Wavelet-Galerkin scheme to solve the elasticity problem associated with each iteration of the topology optimization sequence. Typically, finite element methods are used for this analysis. However, as the mesh size grows, the computational requirements necessary to solve the finite element equations increase beyond the capacity of current desk top computers. This problem is inherent to finite element methods, as the condition number of finite element matrices increases with mesh size. Wavelet-Galerkin techniques are used to replace standard finite element methods in an attempt to alleviate this problem. Examples demonstrating the performance of the new methodology are presented.

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