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Isogeometric Bézier dual mortaring: The biharmonic problem

Di Miao, Michael A. Scott, Michael J. Borden, Derek C. Thomas, Zhihui Zou

TL;DR

The paper develops a dual mortar method to solve the biharmonic problem on multi-patch isogeometric domains by weakly enforcing $C^1$ continuity across interfaces using both global and Bézier dual basis functions. By constructing a sparse, well-posed discretization with dual-based Lagrange multipliers, the approach maintains sparsity of the condensed system and achieves optimal or near-optimal convergence in several benchmark problems, despite Bézier dual basis’s sub-optimal polynomial reproduction. The authors provide a rigorous error analysis, practical discretization strategies (including master/slave patch choices and vertex treatments), and extensive numerical validation on two- and multi-patch geometries as well as a membrane vibration test. They also discuss inf-sup stability considerations, the role of the consistency error, and potential extensions to Kirchhoff-Love shells, outlining avenues for restoring optimal approximation power in future work.

Abstract

In this paper we develop an isogeometric Bézier dual mortar method for the biharmonic problem on multi-patch domains. The well-posedness of the discrete biharmonic problem requires a discretization with $C^1$ continuous basis functions. Hence, two Lagrange multipliers are required to apply both $C^0$ and $C^1$ continuity constraints on each intersection. The dual mortar method utilizes dual basis functions to discretize the Lagrange multiplier spaces. In order to preserve the sparsity of the coupled problem, we develop a dual mortar suitable $C^1$ constraint and utilize the Bézier dual basis to discretize the Lagrange multiplier spaces. The Bézier dual basis functions are constructed through Bézier projection and possess the same support size as the corresponding B-spline basis functions. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. Although the Bézier dual basis is sub-optimal due to the lack of polynomial reproduction, our formulation successfully postpones the domination of the consistency error for practical problems. We verify the theoretical results and demonstrate the performance of the proposed formulation through several benchmark problems.

Isogeometric Bézier dual mortaring: The biharmonic problem

TL;DR

The paper develops a dual mortar method to solve the biharmonic problem on multi-patch isogeometric domains by weakly enforcing continuity across interfaces using both global and Bézier dual basis functions. By constructing a sparse, well-posed discretization with dual-based Lagrange multipliers, the approach maintains sparsity of the condensed system and achieves optimal or near-optimal convergence in several benchmark problems, despite Bézier dual basis’s sub-optimal polynomial reproduction. The authors provide a rigorous error analysis, practical discretization strategies (including master/slave patch choices and vertex treatments), and extensive numerical validation on two- and multi-patch geometries as well as a membrane vibration test. They also discuss inf-sup stability considerations, the role of the consistency error, and potential extensions to Kirchhoff-Love shells, outlining avenues for restoring optimal approximation power in future work.

Abstract

In this paper we develop an isogeometric Bézier dual mortar method for the biharmonic problem on multi-patch domains. The well-posedness of the discrete biharmonic problem requires a discretization with continuous basis functions. Hence, two Lagrange multipliers are required to apply both and continuity constraints on each intersection. The dual mortar method utilizes dual basis functions to discretize the Lagrange multiplier spaces. In order to preserve the sparsity of the coupled problem, we develop a dual mortar suitable constraint and utilize the Bézier dual basis to discretize the Lagrange multiplier spaces. The Bézier dual basis functions are constructed through Bézier projection and possess the same support size as the corresponding B-spline basis functions. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. Although the Bézier dual basis is sub-optimal due to the lack of polynomial reproduction, our formulation successfully postpones the domination of the consistency error for practical problems. We verify the theoretical results and demonstrate the performance of the proposed formulation through several benchmark problems.

Paper Structure

This paper contains 26 sections, 8 theorems, 97 equations, 28 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prior work
  3. Dual basis functions
  4. Global dual basis
  5. Bézier dual basis
  6. The dual mortar method
  7. A dual mortar formulation for the multi-patch biharmonic problem
  8. Domain decomposition
  9. The biharmonic problem
  10. Dual-compatible $C^1$ constraints
  11. The dual mortar formulation
  12. Discretization
  13. Building a basis for the null space of $\mathbf{B}_b$
  14. The two-patch case
  15. The multi-patch case
  16. ...and 11 more sections

Key Result

Lemma 1

Given two differentiable bijective geometric mappings $\mathbf{F}_{s}\colon\hat{\Omega}_s\rightarrow{\Omega}_s$ and $\mathbf{F}_{m}\colon\hat{\Omega}_m\rightarrow{\Omega}_m$, a $C^0$-continuous function $u$ is $C^1$-continuous in the physical domain if and only if

Figures (28)

  • Figure 1: A geometry and two modelling strategies: trimming and reparameterization.
  • Figure 2: A comparison of a B-spline basis function (a) with the corresponding global dual basis function (b) and the Bézier dual basis function (c).
  • Figure 3: An example of a domain decomposition of a triangular domain. The patches are defined on different parametric domains and are connected via geometric mappings.
  • Figure 4: A two-patch planar domain $\Omega$ consisting of two patches $\Omega_s$ and $\Omega_m$ that are defined by the two mappings $\mathbf{F}_s$ and $\mathbf{F}_m$.
  • Figure 5: Two exemplary basis functions in the constrained space $\mathcal{K}_b^h$.
  • ...and 23 more figures

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 11 more