Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain
26 Jul 2017-Annals of the West University of Timisoara: Mathematics and Computer Science (Sciendo)-Vol. 55, Iss: 1, pp 129-147
TL;DR: In this article, an alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general is used to solve the invese problem for the biharmonic equation.
Abstract: Abstract In this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM) for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.
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TL;DR: In this article, an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part of the biharmonic equation, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary.
Abstract: An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.
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TL;DR: In this paper, an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of boundary.
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07 Jan 2021
TL;DR: In this article, a severely ill posed problem associated with a two dimensional homogeneous biharmonic equation was considered and a two-parameter regularization method was proposed to obtain a stable solution which converges to the solution.
Abstract: In this paper we consider a severely ill posed problem associated with a two dimensional homogeneous biharmonic equation. By perturbing the original problem and using a two parameters regularization method, we get a stable solution which converges to the solution of the considered problem. Under some priori bound assumptions, different errors estimates for the regularized solution are obtained. These last ones depend on the choice of the space of the exact solution. To show the effectiveness of the proposed regularization method some numerical results are given.
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01 Jan 2012
139,059 citations
"Iterative Method to Solve a Data Co..." refers methods in this paper
...The KMF iterative algorithm investigated proposed by [10] is a convergent algorithm for solving elliptic Cauchy problems and is widely used in the case of Cauchy problems with Laplace’s and Poisson’s equation [11], [13] and [14] and propsed in this work in the case of the biharmonic problem with the specified conditions in (2....
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...[10], that was widely used and studied [11], [12], [13], [14]....
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Book•
01 Jan 1952
2,399 citations
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28 Feb 1984TL;DR: In this article, the authors propose a method of approximate boundary refinement based on the theory of elasticity, and apply it to two-dimensional problems with different types of boundary conditions.
Abstract: 1 Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definitions.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.4.1. The Collocation Method.- 1.4.2. Method of Collocation by Subregions.- 1.5. Method of Galerkin.- 1.6. Weak Formulations.- 1.7. Inverse Problem and Boundary Solutions.- 1.8. Classification of Approximate Methods.- References.- 2 Potential Problems.- 2.1. Introduction.- 2.2. Elements of Potential Theory.- 2.3. Indirect Formulation.- 2.4. Direct Formulation.- 2.5. Boundary Element Method.- 2.6. Two-Dimensional Problems.- 2.6.1. Source Formulation.- 2.7. Poisson Equation.- 2.8. Subregions.- 2.9. Orthotropy and Anisotropy.- 2.10. Infinite Regions.- 2.11. Special Fundamental Solutions.- 2.12. Three-Dimensional Problems.- 2.13. Axisymmetric Problems.- 2.14. Axisymmetric Problems with Arbitrary Boundary Conditions.- 2.15. Nonlinear Materials and Boundary Conditions.- 2.15.1. Nonlinear Boundary Conditions.- References.- 3 Interpolation Functions.- 3.1. Introduction.- 3.2. Linear Elements for Two-Dimensional Problems.- 3.3. Quadratic and Higher-Order Elements.- 3.4. Boundary Elements for Three-Dimensional Problems.- 3.4.1. Quadrilateral Elements.- 3.4.2. Higher-Order Quadrilateral Elements.- 3.4.3. Lagrangian Quadrilateral Elements.- 3.4.4. Triangular Elements.- 3.4.5. Higher-Order Triangular Elements.- 3.5. Three-Dimensional Cell Elements.- 3.5.1. Tetrahedron.- 3.5.2. Cube.- 3.6. Discontinuous Boundary Elements.- 3.7. Order of Interpolation Functions.- References.- 4 Diffusion Problems.- 4.1. Introduction.- 4.2. Laplace Transforms.- 4.3. Coupled Boundary Element - Finite Difference Methods.- 4.4. Time-Dependent Fundamental Solutions.- 4.5. Two-Dimensional Problems.- 4.5.1. Constant Time Interpolation.- 4.5.2. Linear Time Interpolation.- 4.5.3. Quadratic Time Interpolation.- 4.5.4. Space Integration.- 4.6. Time-Marching Schemes.- 4.7. Three-Dimensional Problems.- 4.8. Axisymmetric Problems.- 4.9. Nonlinear Diffusion.- References.- 5 Elastostatics.- 5.1. Introduction to the Theory of Elasticity.- 5.1.1. Initial Stresses or Initial Strains.- 5.2. Fundamental Integral Statement.- 5.2.1. Somigliana Identity.- 5.3. Fundamental Solutions.- 5.4. Stresses at Internal Points.- 5.5. Boundary Integral Equation.- 5.6. Infinite and Semi-Infinite Regions.- 5.7. Numerical Implementation.- 5.8. Boundary Elements.- 5.9. System of Equations.- 5.10. Stresses and Displacements Inside the Body.- 5.11. Stresses on the Boundary.- 5.12. Surface Traction Discontinuities.- 5.13. Two-Dimensional Elasticity.- 5.14. Body Forces.- 5.14.1. Gravitational Loads.- 5.14.2. Centrifugal Load.- 5.14.3. Thermal Loading.- 5.15. Axisymmetric Problems.- 5.15.1. Extension to Nonaxisymmetric Boundary Values.- 5.16. Anisotropy.- References.- 6 Boundary Integral Formulation for Inelastic Problems.- 6.1. Introduction.- 6.2. Inelastic Behavior of Materials.- 6.3. Governing Equations.- 6.4. Boundary Integral Formulation.- 6.5. Internal Stresses.- 6.6. Alternative Boundary Element Formulations.- 6.6.1. Initial Strain.- 6.6.2. Initial Stress.- 6.6.3. Fictitious Tractions and Body Forces.- 6.7. Half-Plane Formulations.- 6.8. Spatial Discretization.- 6.9. Internal Cells.- 6.10. Axisymmetric Case.- References.- 7 Elastoplasticity.- 7.1. Introduction.- 7.2. Some Simple Elastoplastic Relations.- 7.3. Initial Strain: Numerical Solution Technique.- 7.3.1. Examples - Initial Strain Formulation.- 7.4. General Elastoplastic Stress-Strain Relations.- 7.5. Initial Stress: Outline of Solution Techniques.- 7.5.1. Examples: Kelvin Implementation.- 7.5.2. Examples: Half-Plane Implementation.- 7.6. Comparison with Finite Elements.- References.- 8 Other Nonlinear Material Problems.- 8.1. Introduction.- 8.2. Rate-Dependent Constitutive Equations.- 8.3. Solution Technique: Viscoplasticity.- 8.4. Examples: Time-Dependent Problems.- 8.5. No-Tension Materials.- References.- 9 Plate Bending.- 9.1. Introduction.- 9.2. Governing Equations.- 9.3. Integral Equations.- 9.3.1. Other Fundamental Solutions.- 9.4. Applications.- References.- 10 Wave Propagation Problems.- 10.1. Introduction.- 10.2. Three-Dimensional Water Wave Propagation Problems.- 10.3. Vertical Axisymmetric Bodies.- 10.4. Horizontal Cylinders of Arbitrary Section.- 10.5. Vertical Cylinders of Arbitrary Section.- 10.6. Transient Scalar Wave Equation.- 10.7. Three-Dimensional Problems: The Retarded Potential.- 10.8. Two-Dimensional Problems.- References.- 11 Vibrations.- 11.1. Introduction.- 11.2. Governing Equations.- 11.3. Time-Dependent Integral Formulation.- 11.4. Laplace Transform Formulation.- 11.5. Steady-State Elastodynamics.- 11.6. Free Vibrations.- References.- 12 Further Applications in Fluid Mechanics.- 12.1. Introduction.- 12.2. Transient Groundwater Flow.- 12.3. Moving Interface Problems.- 12.4. Axisymmetric Bodies in Cross Flow.- 12.5. Slow Viscous Flow (Stokes Flow).- 12.6. General Viscous Flow.- 12.6.1. Steady Problems.- 12.6.2. Transient Problems.- References.- 13 Coupling of Boundary Elements with Other Methods.- 13.1. Introduction.- 13.2. Coupling of Finite Element and Boundary Element Solutions.- 13.2.1. The Energy Approach.- 13.3. Alternative Approach.- 13.4. Internal Fluid Problems.- 13.4.1. Free-Surface Boundary Condition.- 13.4.2. Extension to Compressible Fluid.- 13.5. Approximate Boundary Elements.- 13.6. Approximate Finite Elements.- References.- 14 Computer Program for Two-Dimensional Elastostatics.- 14.1. Introduction.- 14.2. Main Program and Data Structure.- 14.3. Subroutine INPUT.- 14.4. Subroutine MATRX.- 14.5. Subroutine FUNC.- 14.6. Subroutine SLNPD.- 14.7. Subroutine OUTPT.- 14.8. Subroutine FENC.- 14.9. Examples.- 14.9.1. Square Plate.- 14.9.2. Cylindrical Cavity Problem.- References.- Appendix A Numerical Integration Formulas.- A.1. Introduction.- A.2. Standard Gaussian Quadrature.- A.2.1. One-Dimensional Quadrature.- A.2.2. Two- and Three-Dimensional Quadrature for Rectangles and Rectangular Hexahedra.- A.2.3. Triangular Domain.- A.3. Computation of Singular Integrals.- A.3.1. One-Dimensional Logarithmic Gaussian Quadrature Formulas.- A.3.3. Numerical Evaluation of Cauchy Principal Values.- References.- Appendix B Semi-Infinite Fundamental Solutions.- B.1. Half-Space.- B.2. Half-Plane.- References.- Appendix C Some Particular Expressions for Two-Dimensional Inelastic Problems.
1,424 citations
"Iterative Method to Solve a Data Co..." refers background or methods in this paper
...differential equations over domain discretisation methods, such as finite difference or finite element method, are well-known [15]....
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...The boundary integral equations are classical kind of formulations for partial diferential equations [15], [17]....
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587 citations
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...Thus; we consider that the solution u is in the space H20(4(2),Ω), defined by [19]: H(2) 0(4(2),Ω) = {u ∈ H(2)(Ω) / 4(2)u = 0} (2....
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01 Jan 1991
270 citations