A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals

Ken Hayami

ISBN 10: 3540550003 / ISBN 13: 9783540550006

Editore: Springer Berlin Heidelberg, Springer Berlin Heidelberg Mär 1992, 1992

Riguardo questo articolo

This item is printed on demand - Print on Demand Titel. Neuware -In three dimensional boundary element analysis, computation of integrals is an important aspect since it governs the accuracy of the analysis and also because it usually takes the major part of the CPU time. The integrals which determine the influence matrices, the internal field and its gradients contain (nearly) singular kernels of order lIr a (0:= 1,2,3,4, ) where r is the distance between the source point and the integration point on the boundary element. For planar elements, analytical integration may be possible 1,2,6. However, it is becoming increasingly important in practical boundary element codes to use curved elements, such as the isoparametric elements, to model general curved surfaces. Since analytical integration is not possible for general isoparametric curved elements, one has to rely on numerical integration. When the distance d between the source point and the element over which the integration is performed is sufficiently large compared to the element size (d> 1), the standard Gauss-Legendre quadrature formula 1,3 works efficiently. However, when the source is actually on the element (d=O), the kernel 1I~ becomes singular and the straight forward application of the Gauss-Legendre quadrature formula breaks down. These integrals will be called singular integrals. Singular integrals occur when calculating the diagonals of the influence matrices.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 468 pp. Englisch.

Codice articolo 9783540550006

Dati bibliografici

• Titolo: A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals
• Editore: Springer Berlin Heidelberg, Springer Berlin Heidelberg Mär 1992
• Data di pubblicazione: 1992
• Lingua: Inglese
• ISBN 10: 3540550003
• ISBN 13: 9783540550006
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 242 L x 170 A x 26 P (millimetri) 242 millimetri di larghezza per 170 millimetri di altezza per 26 millimetri di profondità
• Peso: 794 grammi

Descrizione articolo

Riassunto

This book proposes an accurate and efficient numerical inte- gration method for nearly singular integrals over general curved surfaces, arising in threedimensional boundary ele- ment analysis. Nearly singular integrals frequently occur in engineering problems involving thin structures or gaps and when calculating the field near the boundary. Numerical ex- periments show that the method is far more efficient compa- red to previous methods and is robust concerning the type of integral kernel and position of the source point. Theoreti- cal error estimates for the method is derived using complex function theory. The method is also shown to be applicable to weakly singular and hypersingular integrals. Knowledge in basic calculus is assumed. The book is intended for engine- ers and researchers using the boundary element method who require accurate methods for numerical integration and also for numerical analysts interested in a rich application area for numerical integration.

Contenuti

I Theory and Algorithms.- 1 Introduction.- 2 Boundary Element Formulation of 3-d Potential Problems.- 2.1 Boundary Integral Equation.- 2.2 Treatment of the Exterior Problem.- 2.3 Discretization into Boundary Elements.- 2.4 Row Sum Elimination Method.- 3 Nature of Integrals in 3-d Potential Problems.- 3.1 Weakly Singular Integrals.- 3.2 Hyper Singular Integrals.- 3.3 Nearly Singular Integrals.- 4 Survey of Quadrature Methods for 3-d Boundary Element Method.- 4.1 Closed Form Integrals.- 4.2 Gaussian Quadrature Formula.- 4.3 Quadrature Methods for Singular Integrals.- (1) The weighted Gauss method.- (2) Singularity subtraction and Taylor expansion method.- (3) Variable transformation methods.- (4) Coordinate transformation methods.- 4.4 Quadrature Methods for Nearly Singular Integrals.- (1) Element subdivision.- (2) Variable transformation methods.- (3) Polar coordinates.- 5 The Projection and Angular & Radial Transformation (Part) Method.- 5.1 Introduction.- 5.2 Source Projection.- 5.3 Approximate Projection of the Curved Element.- 5.4 Polar Coordinates in the Projected Element.- 5.5 Radial Variable Transformation.- (i) Weakly Singular Integrals.- (ii) Nearly Singular Integrals.- (1) Singularity cancelling radial variable transformation.- (2) Consideration of exact inverse projection and curvature of the element in the transformation.- (3) Adaptive logarithmic transformation (log-L2).- (4) Adaptive logarithmic transformation (log-L1).- (5) Single and double exponential transformations.- 5.6 Angular Variable Transformation.- (1) Adaptive logarithmic angular variable transformation.- (2) Single and double exponential transformations.- 5.7 Implementation of the PART method.- (1) The use of Gauss-Legendre formula.- (2) The use of truncated trapezium rule.- 5.8 Variable Transformation in the Parametric Space.- 6 Elementary Error Analysis.- 6.1 The Use of Error Estimate for Gauss-Legendre Quadrature Formula.- 6.2 Case ß = 2 (Adaptive Logarithmic Transformation: log-L2).- 6.3 Case ß = l Transformation.- 6.4 Case ß = 3 Transformation.- 6.5 Case ß = 4 Transformation.- 6.6 Case ß = 5 Transformation.- 6.7 Summary of Error Estimates for ß=1~5.- 6.8 Error Analysis for Flux Calculations.- 7 Error Analysis using Complex Function Theory.- 7.1 Basic Theorem.- 7.2 Asymptotic Expression for the Error Characteristic Function ?n(z).- 7.3 Use of the Elliptic Contour as the Integral Path.- 7.4 The Saddle Point Method.- 7.5 Integration in the Transformed Radial Variable: R.- 7.6 Error Analysis for the Identity Transformation: R(?) = ?.- (1) Estimation of the size ? of the ellipse ??.- (2) Estimation of max | f(z) | z???.- (3) Error estimate En(f).- 7.7 Error Analysis for the log-L2 Transformation.- (1) Case: ? = odd.- (2) Case: ? = even.- (i) Contribution from the branch line l+, l-.- (ii) Contribution from the ellipse ??.- (iii) Contribution from the small circle C?.- (iv) Summary.- 7.8 Error Analysis for the log-L1 Transformation.- (1) Error analysis using the saddle point method.- (2) Error analysis using the elliptic contour: ??.- (i) Estimation of max | f(z) | z???.- (ii) Estimation of ?.- (iii) Error estimate En(f).- 7.9 Summary of Theoretical Error Estimates.- II Applications and Numerical Results.- 8 Numerical Experiment Procedures and Element Types.- 8.1 Notes on Procedures for Numerical Experiments.- 8.2 Geometry of Boundary Elements used for Numerical Experiments.- (1) Planar rectangle (PLR).- (2) ‘Spherical’ quadrilateral (SPQ).- (3) Hyperbolic quadrilateral (HYQ).- 9 Applications to Weakly Singular Integrals.- 9.1 Check with Analytial Integration Formula for Constant Planar Elements.- 9.2 Planar Rectangular Element with Interpolation Function ?ij.- 9.3 ‘Spherical’ Quadrilateral Element with Interpolation Function ?ij.- (1) Results for ?S?iju* dS.- (2) Results for ?S?ijq* dS.- 9.4 Hyperbolic Quadrilateral Element with Interpolation Function ?ij.- (1) Results for ?s?iju* dS.- (2) Results for ?s?ijq*dS.- 9.5 Summary of Numerical Results for Weakly Singular Integrals.- 10 Applications to Nearly Singular Integrals.- 10.1 Analytical Integration Formula for Constant Planar Elements.- 10.2 Singularity Cancelling Radial Variable Transformation for Constant Planar Elements.- 10.3 Application of the Singularity Cancelling Transformation to Elements with Curvature and Interpolation Functions.- (1) Application to curved elements.- (2) Application to integrals including interpolation functions.- 10.4 The Derivation of the log-L2 Radial Variable Transformation.- (1) Application of radial variable transformations ?d? = r’?dR (ß? ?) to integrals ?s 1/r?dS over curved elements.- (2) Difficulty with flux calculation.- 10.5 The log-L1 Radial Variable Transformation.- 10.6 Comparison of Radial Variable transformations for the Model Radial Integral I?,?.- (1) Transformation based on the Gauss-Legendre rule.- (i) Identity transformation.- (ii) log-L2 transformation.- (iii) log-L1 transformation.- (2) Transformation based on the truncated trapezium rule.- (i) Single Exponential (SE) transformation.- (ii) Double Exponential (DE) transformation.- (3) Summary.- 10.7 Comparison of Different Numerical Integration methods on the ‘spherical’ Element.- (1) Effect of the source distance d.- (2) Effect of the position of the source projection xs.- 10.8 Summary of Numerical Results for Nearly Singular Integrals.- 11 Applications to Hypersingular Integrals.- 12 Conclusions.- References.