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Nitsche's method for two and three dimensional NURBS patch coupling

Vinh Phu Nguyen, Pierre Kerfriden, Marco Brino, Stephane Bordas, Elvio Bonisoli

arXiv:1308.0802v1math.NA

TL;DR

Non-conforming multi-patch NURBS geometries create an interface-coupling challenge for IGA. The paper develops a weak Nitsche method for two- and three-dimensional patch coupling and demonstrates it in static linear-elastic analyses. The authors conclude that the method enlarges the applicability of NURBS-based IGA, while noting that dynamics and nonlinear materials remain unresolved extensions.

  • Problem

    Multiple-patch NURBS are common for complex geometries, but conforming interface parametrizations are virtually impossible, limiting multi-patch IGA coupling.

  • Method

    The paper couples non-conforming NURBS patches weakly with Nitsche’s method, using interface stabilization and independently discretized domains.

  • Results

    Numerical examples demonstrate good performance for two- and three-dimensional static linear-elastic stress analyses, including multi-interface connecting-rod coupling.

  • Takeaways & Limitations

    The proposed coupling method enlarges the applicability of NURBS-based isogeometric analysis to non-conforming multi-patch geometries.

  • Takeaways & Limitations

    The contribution is limited to linear elastostatic problems; extensions to dynamics and nonlinear materials remain under investigation.

Abstract

from arXiv · show

A Nitche's method is presented to couple non-conforming two and three dimensional NURBS (Non Uniform Rational B-splines) patches in the context of isogeometric analysis (IGA). We present results for elastic stress analyses under the static condition of two and three dimensional NURBS geometries. The contribution fills the gap in the literature and enlarges the applicability of NURBS-based isogeometric analysis.

1. Introduction

IGA uses NURBS as both geometry and analysis discretizations, but non-conforming multi-patch interfaces remain a central coupling challenge. The paper addresses this with a weak Nitsche coupling for two- and three-dimensional NURBS patches.

  • NURBS and IGA: NURBS represent complex CAD geometries exactly, including conic and circular sections that polynomial functions only approximate.Their established evaluation and refinement algorithms support analysis-oriented use in IGA.
  • NURBS and IGA: IGA extends NURBS from geometry representation to analysis discretization across structural, solid, fluid, and contact mechanics.The approach has also been applied to optimization, boundary elements, higher-order PDEs, vibration, and fracture.
  • Applications: Structural mechanics benefits from smooth NURBS bases through straightforward plate and shell construction, rotation-free thin-shell formulations, and reduced shear-locking.These benefits are reported relative to conventional approaches.
  • Applications: Smooth NURBS contact surfaces can produce more physically accurate contact stresses than faceted conventional geometry discretizations.Faceted surfaces may cause traction jumps and oscillations unless very fine meshes are used.
  • Multi-patch coupling: Complex-topology objects are usually represented as multiple-patch NURBS, but conforming parametrizations at patch interfaces are virtually impossible.This makes high-continuity multi-patch coupling an important IGA research topic.
  • Multi-patch coupling: The paper presents Nitsche’s method to couple non-conforming two- and three-dimensional NURBS patches weakly.The method offers an alternative to same-parametrization constraints and avoids relying on T-splines, which are not yet standard in CAD.

2. Problem description, governing equations and weak form

The paper formulates linear elastostatic coupling across an internal interface separating independently represented NURBS patches. Its weak formulation uses Nitsche terms, jump and average operators, and stabilization to handle nonmatching discretizations.

  • 2.1. Governing equations: The domain is divided into two non-overlapping subdomains, each representing a NURBS patch, across one internal boundary Γ*.The external boundary is separated into Dirichlet and Neumann portions.
  • 2.1. Governing equations: The primary unknown is displacement, while stress, strain, constitutive tensors, prescribed data, and outward normals define the linear elastostatic problem.The formulation assumes small strain and includes linear isotropic elasticity through Young’s modulus, Poisson’s ratio, and Lamé constants.
  • 2.2. Weak form: Solution and trial spaces are defined independently over each subdomain before constructing the coupled weak formulation.The spaces contain the solution and trial functions, respectively.
  • 2.2. Weak form: Nitsche coupling seeks the pair of patchwise displacements in the product solution space such that the interface weak form holds for all paired test functions.Essential boundary conditions are enforced pointwise where possible or by methods other than Nitsche’s method.
  • 2.2. Weak form: The interface formulation uses jump and average operators, with a generalized stress average controlled by 0 ≤ γ ≤ 1.γ = 0.5 gives the usual average, while γ = 0 or 1 yields one-sided mortaring; the standard average is used unless stated otherwise.
  • 2.2. Weak form: The stabilization parameter α distinguishes Nitsche’s method from the penalty method and can be chosen above a minimum ensuring positive definiteness of the bilinear form.The method otherwise shares the penalty method’s form except for its second and third left-hand-side terms.
  • 3.3. Discrete equations: For discretization, the weak form is rewritten in matrix form with independently discretized subdomains and an interface mesh mismatch.The displacement approximation is then introduced for the two domains.
  • 2.2. Weak form: Voigt notation represents stress and strain as column vectors, with separate matrix forms specified for two and three dimensions.The notation uses engineering shear-strain components in the strain vector.

3. Discretisation

The discretisation uses NURBS bases to represent geometry and fields, with mappings from parent through parametric to physical space. Independently discretised domains are coupled across mismatched interfaces through discrete bulk and interfacial matrices.

  • 3.1. NURBS: Open knot vectors and Cox-de Boor recursion define B-spline bases with partition of unity, nonnegative values, local support, and knot-dependent continuity.Repeated knots reduce continuity, while open knots provide interpolatory behavior at interval ends.
  • 3.1. NURBS: NURBS bases combine B-spline functions with positive weights, representing curves including polynomials and circular arcs; equal weights recover the B-spline basis.Weights for simple geometries may be defined analytically, while complex-geometry weights can come from CAD software.
  • 3.2. Isogeometric analysis: IGA uses CAD-generated basis functions to discretise both geometry and unknown fields, with element domains defined by regions between knot lines having non-zero parametric area.The field approximation uses the same type of basis expansion as the geometry, although its coefficients are generally non-interpolatory at nodes.
  • 3.2. Isogeometric analysis: NURBS elements require mappings from parent space to parametric space and then physical space, with the physical mapping given by their composition.Parent space is used for numerical quadrature, and the element geometry is expressed through local NURBS basis functions and control points.
  • 3.3. Discrete equations: The two domains are discretised independently, creating an interface mesh mismatch that is handled using bulk stiffness and interfacial coupling matrices.The discrete system also includes the external force vector, while stresses, strains, and displacements follow standard finite-element relations.
  • 3.3. Discrete equations: For hierarchical meshes, the interface is integrated using fine elements from one domain, with Gauss points mapped to the intersecting elements of the other domain.The interface discretisation uses element edges of the fine mesh that intersect the interface.

4. Implementation

The implementation evaluates Nitsche coupling integrals on interface trace meshes, maps quadrature points between physical, parent, and NURBS parameter spaces, and assembles the resulting matrices globally for 2D and 3D problems.

  • General implementation: For standard finite elements, coupling-matrix implementation focuses on interface integrals after independent domain discretisation, while bulk stiffness assembly remains standard.The presentation first treats Lagrange elements and then generalises to NURBS with minor modifications.
  • Hierarchical meshes: Hierarchical meshes use fine interface elements to evaluate integrals, with the corresponding coarse-side element identified from interface location.This known element correspondence makes hierarchical meshes straightforward to integrate.
  • Hierarchical meshes: Interface Gauss points are mapped to global coordinates and then to the parent elements of both coupled domains to evaluate shape functions and coupling terms.The mapping uses line-element shape functions for the interface and element shape functions for each adjoining domain.
  • Non-matching structured meshes: Non-matching structured meshes use the trace mesh of Ω1 on Γ* for numerical integration and data structures recording each Gauss point’s containing elements.The resulting Gauss points are passed to the coupling-matrix assembly procedure.
  • Three-dimensional implementation: In 3D, each trace-face Gauss point is transformed to physical space, weighted using tangent and normal information, and associated with an element in Ω2.The procedure computes tangent vectors, a normal vector, and the transformed integration weight before parent-space mappings.
  • NURBS implementation: For NURBS, parent-space quadrature points are first mapped to parameter space with Jacobian-adjusted weights, after which global NURBS mappings and FindSpan identify elements.Bézier extraction can avoid these parameter-space modifications and facilitate integration into existing finite-element codes.

5. Numerical examples

Three examples assess the proposed coupling method from a 2D Timoshenko beam through 3D cantilever and connecting-rod geometries. The reported solutions agree with exact or standard Galerkin results and exhibit optimal convergence in the beam study.

  • Study design: The numerical study increases in complexity from 2D/2D Timoshenko-beam coupling to 3D/3D cantilever and connecting-rod coupling.The first two examples verify implementation, while the Timoshenko beam also receives convergence analysis.
  • Timoshenko beam: Timoshenko-beam midline displacements show good agreement with the exact solution, and stresses are reported for conforming and non-conforming discretisations.The study uses bilinear Lagrange elements and higher-order B-spline elements under plane stress.
  • Timoshenko beam: Non-hierarchical bi-cubic meshes produce a quadratic stress profile with the theoretical maximum value of 250 along the beam midline.The left domain uses an 8 × 6 mesh and the right domain a 4 × 3 mesh.
  • Timoshenko beam: The Timoshenko-beam convergence plots show optimal convergence rates for both displacement and energy norms.The stabilisation parameter is computed per mesh using a polynomial-order-dependent rule chosen to keep the stiffness matrix positive definite.
  • 3D cantilever beam: For the 3D cantilever, the non-conforming B-spline solution’s σxx contour shows good agreement with a standard Galerkin discretisation.The comparison uses 32 × 4 × 4 tri-cubic B-spline elements for the Galerkin model.
  • Connecting rod: The connecting-rod example couples three NURBS patches across two interfaces and compares displacement and stress fields with a Siemens-NX finite-element model.The IGA model contains 4224 elements and 11305 control points; the reported distributions show the expected Saint-Venant displacement pattern and three-dimensional stress effects.

6. Conclusions

The paper presents and implements Nitsche coupling for non-conforming NURBS patches, with numerical examples demonstrating good performance. Its demonstrated scope is limited to linear elastostatic problems.

  • Contributions: The paper presents a Nitsche method for coupling non-conforming NURBS patches and provides detailed implementation and numerical demonstrations.The stated aim is to enlarge the applicability of NURBS-based isogeometric analysis.
  • Scope: The contribution is limited to linear elastostatic problems, while extensions to dynamics and nonlinear materials remain under investigation.The paper does not claim viability for those broader multi-patch settings yet.
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