PaperlayerView exact revision on arXiv

Source-linked AI summary

An Isogeometric Boundary Element Method for elastostatic analysis: 2D implementation aspects

R. N. Simpson, S. P. A. Bordas, H. Lian, J. Trevelyan

arXiv:1302.5305v1math.NA

TL;DR

The paper addresses the mismatch between CAD design representations and analysis discretisations. It presents a 2D IGABEM implementation that uses CAD-based NURBS geometry and fields, and reports good FEM agreement with improved accuracy over standard quadratic BEM. The method retains exact geometry during refinement, but its coefficients do not necessarily represent physical displacements and tractions.

  • Problem

    Conventional analysis requires creating a separate analysis-ready boundary discretisation from CAD, contributing to the gap between engineering design and analysis.

  • Method

    The paper implements 2D elastostatic IGABEM by using CAD NURBS basis functions for geometry, displacement, and traction approximations, with Greville abscissae for boundary collocation and knot insertion or order elevation for refinement.

  • Results

    The example shows good agreement with FEM and significant accuracy improvements over standard BEM using quadratic basis functions.

  • Takeaways & Limitations

    IGABEM can use CAD data directly for analysis without creating a separate mesh while supporting richer basis functions through refinement.

  • Takeaways & Limitations

    NURBS coefficients do not necessarily represent real displacements and tractions because NURBS basis functions need not satisfy the Kronecker-delta property.

Abstract

from arXiv · show

The concept of isogeometric analysis, whereby the parametric func- tions that are used to describe CAD geometry are also used to approx- imate the unknown fields in a numerical discretisation, has progressed rapidly in recent years. This paper advances the field further by outlin- ing an isogeometric Boundary Element Method (IGABEM) that only re- quires a representation of the geometry of the domain for analysis, fitting neatly with the boundary representation provided completely by CAD. The method circumvents the requirement to generate a boundary mesh representing a significant step in reducing the gap between engineering design and analysis. The current paper focuses on implementation details of 2D IGABEM for elastostatic analysis with particular attention paid towards the differences over conventional boundary element implementa- tions. Examples of Matlab R° code are given whenever possible to aid understanding of the techniques used.

1. Introduction

Engineering design models must be converted from CAD into analysis-ready discretisations, creating a time-consuming and specialist-dependent bottleneck. Isogeometric analysis addresses this mismatch by using CAD parametrisations for numerical approximation, while IGABEM extends the idea to boundary analysis using only a surface representation.

  • Design–analysis gap: Creating an analysis-ready discretisation from CAD is often the most time-consuming design-to-analysis step and may require specialist intervention for sufficient accuracy.The discretisation forms the computational representation of the domain or boundary.
  • Design–analysis gap: Design and analysis are connected through an iterative process because analysis results can alter the original design.
  • Isogeometric analysis: Isogeometric analysis uses CAD parametric functions, commonly NURBS, to approximate both geometry and unknown fields instead of conventional piecewise polynomial shape functions.
  • Isogeometric analysis: NURBS-based IGA has shortcomings in producing watertight geometries and supporting local refinement, motivating developments such as T-splines and PHT-splines.
  • Paper focus: This paper develops implementation details for 2D isogeometric BEM, building from an initial CAD model to the final BEM system while requiring only a surface representation of the geometry.

2. Geometrical modelling

B-splines and NURBS provide the parametric CAD functions used to represent geometry and approximate unknown fields in isogeometric analysis. The section introduces their control points, knot vectors, basis functions, and role in IGABEM.

  • B-splines and NURBS: B-splines and NURBS are parametric functions defined through independent parameters, with ξ used as the coordinate in parameter space.
  • B-splines and NURBS: Open knot vectors contain p+1 repeated components at both ends, making the curve interpolatory at its beginning and end.
  • B-splines and NURBS: Repeated knot values reduce curve continuity at the corresponding parameter location, while scaling all knot values preserves their relative structure.
  • B-splines and NURBS: A B-spline requires a curve degree, control points, and a knot vector that determines its parametric structure.
  • B-spline and NURBS basis functions: B-spline basis functions are defined recursively and can be evaluated efficiently using algorithms such as Cox-de-Boor and extraction operators.
  • B-spline and NURBS basis functions: NURBS extend B-splines with weighting coordinates; setting all weights to unity recovers B-spline basis functions, so later examples use NURBS for generality.
  • B-spline and NURBS basis functions: In isogeometric analysis, the CAD basis functions approximate displacement and traction fields, allowing CAD to provide the boundary discretisation for BEM.

3. Conventional BEM

Conventional BEM discretizes the boundary integral equation with piecewise polynomial shape functions, collocates equations at nodal points, and assembles a linear system relating displacement and traction degrees of freedom.

  • 3. Conventional BEM: The direct-collocation BEM defines source and field points and relates boundary displacements and tractions through the displacement boundary integral equation.The source point is the collocation point, while the field point is sampled during numerical integration.
  • 3. Conventional BEM: The continuous boundary is split into elements, where geometry, displacement, and traction fields are approximated with local polynomial basis functions.Quadratic elements use three local basis functions, illustrated by the continuous quadratic functions in Figure 7.
  • 3. Conventional BEM: Element contributions are integrated using the Jacobian mapping from local coordinates to the boundary and then assembled over all elements.The Jacobian J_e(η) maps local element coordinates to the boundary.
  • 3. Conventional BEM: Collocation at each nodal point produces matrices H and G relating nodal displacements and tractions, which become a linear system after boundary conditions are imposed.H contains T_ij-kernel integrals and jump terms, while G contains U_ij-kernel integrals.
  • 3. Conventional BEM: The resulting matrix A is full and non-symmetric, and the unknown vector can be solved using conventional or fast linear solvers.The system is formed after rearranging the equations according to prescribed displacement and traction boundary conditions.

4. Isogeometric BEM

The 2D IGABEM implementation replaces conventional polynomial basis functions with NURBS and uses CAD-defined geometry, requiring revised collocation and element definitions for numerical integration.

  • 4. Isogeometric BEM: IGABEM uses NURBS basis functions instead of conventional polynomial functions, creating implementation consequences because control points may not lie on the boundary.This makes conventional collocation at nodal positions invalid in general.
  • 4. Isogeometric BEM: The reactor-vessel example represents the exact 2D boundary from CAD using quadratic NURBS basis functions.The example exploits symmetry and uses p = 2 over the entire boundary.
  • 4.1. Element definition: IGABEM defines numerical-integration elements from unique knot-vector values, with each element covering the boundary portion where its relevant basis functions are non-zero.Here, elements are integration constructs rather than conventional physical discretization entities.
  • 4.1. Element definition: For the reactor problem, the first element spans ξ ∈[0, 1] and contains the NURBS basis functions that are non-zero over that interval.The corresponding basis functions are shown in Figure 10.
  • 4.1. Element definition: Gauss-Legendre integration requires mapping each knot-span element from parameter space to parent coordinates over [−1, 1] using a chain-rule Jacobian.The transformation combines the mapping between physical and parameter coordinates with the mapping to local integration coordinates.

Jacobian of transformation:

IGABEM changes conventional BEM primarily through geometry-based connectivity, non-nodal collocation, refinement that preserves exact geometry, and specialized treatment of boundary integrals. The reactor example uses these mechanisms to obtain agreement with FEM and improved accuracy over standard quadratic BEM.

  • Element connectivity: IGABEM approximates geometry, displacement, and traction using local NURBS basis functions connected to global basis functions through element connectivity.The associated coefficients need not equal physical displacements or tractions because NURBS basis functions are not generally interpolatory.
  • Collocation points: Collocation points are not generally control points because curved-boundary control points may lie away from the physical boundary.Their physical coordinates and element boundaries are obtained from the NURBS geometry representation.
  • Refinement: Knot insertion and order elevation enrich the basis while preserving the exact geometry, addressing cases where the original basis cannot capture unknown-field gradients.Order elevation increases quadratic basis functions to cubic functions, while knot insertion adds knots uniformly in the reactor example.
  • Boundary integration: BEM boundary integrals are classified as regular, nearly singular, or singular according to the collocation point’s position relative to the field element.The singular cases include strongly singular T_ij kernels with O(1/r) behavior and weakly singular U_ij kernels with O(ln(1/r)) behavior in 2D.
  • Boundary integration: The implementation treats regular and nearly singular integrals similarly, uses singularity subtraction for strongly singular integrals, and applies the Telles transformation to weakly singular integrals.Singularity subtraction separates regular and singular parts, while the Telles transformation cancels the weak singularity and leaves a regular integrand.
  • Reactor example: The reactor IGABEM analysis shows excellent agreement with FEM and significant accuracy improvement over standard BEM with quadratic basis functions.The comparison uses h-refinement and evaluates the L2 displacement norm for each mesh.

5. Conclusions

The paper concludes that IGABEM can use CAD geometry directly, avoid boundary-mesh generation, and support refinement through richer basis functions. A reactor example demonstrates good FEM agreement and improved accuracy over standard quadratic BEM.

  • 5. Conclusions: IGABEM uses CAD-provided geometry directly without requiring creation of a separate boundary mesh.This is presented as an attractive feature for reducing the gap between engineering design and analysis.
  • 5. Conclusions: Refinement schemes provide additional refinement or richer basis functions in required areas.The paper discusses refinement as part of the implementation framework.
  • 5. Conclusions: The reactor example obtains good agreement with FEM and significant accuracy improvements over standard BEM using quadratic basis functions.These outcomes are reported as demonstrations of the method’s simplicity and accuracy.

Appendix A.1. B-spline derivatives

The appendix describes derivative evaluation for B-spline basis functions by differentiating recursive relations and expressing higher-order derivatives through lower-order basis functions.

  • Appendix A.1. B-spline derivatives: The first-order derivative of a B-spline basis function is expressed using the polynomial order and basis-function index.Higher-order derivatives are obtained by differentiating the defining relation.
  • Appendix A.1. B-spline derivatives: Higher-order B-spline derivatives can be expressed using lower-order basis functions with shifted indices.The resulting representation uses basis functions such as N_{a,p-k} through N_{a+k,p-k}.

Appendix A.2. NURBS derivatives

The appendix extends derivative evaluation from B-splines to NURBS and supplies the elasticity quantities and geometric normal definitions needed by the 2D IGABEM formulation.

  • Appendix A.2. NURBS derivatives: The first-order derivative of a NURBS basis function is derived from the NURBS definition after introducing convenient notation.The appendix then expresses higher-order derivatives in terms of lower-order derivatives.
  • Appendix A.2. NURBS derivatives: The 2D elasticity formulation defines the shear modulus, Poisson’s ratio, and Kronecker-delta function before giving its fundamental solutions.A comma denotes differentiation in the displayed elasticity expressions.
  • Appendix A.2. NURBS derivatives: Boundary normals are calculated from the geometry, with ξ_s and ξ_f denoting the beginning and end knot values of an element.The definition assumes an outward-pointing normal.

Appendix C.1. Control points and weights

This appendix identifies control point coordinates in pascals and the global basis index associated with (b1,b2,b3).

  • Control point coordinates are represented in pascals (Pa).
  • The global basis index is denoted by (a) for (b1,b2,b3).
Loading 1302.5305v1…