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Mixed Variational Formulation of Coupled Plates

Jun Hu, Zhen Liu, Rui Ma, Ruishu Wang

Abstract

This paper proposes a mixed variational formulation for the problem of two coupled plates with a rigid {junction}. The proposed mixed {formulation} introduces {the union of} stresses and moments as {an auxiliary variable}, which {are} commonly of great interest in practical applications. The primary challenge lies in determining a suitable {space involving} both boundary and junction conditions of the auxiliary variable. The {theory} of densely defined operators in Hilbert spaces is employed to define {a nonstandard Sobolev space} without the use of trace operators. The well-posedness is established for the mixed formulation. Based on these conditions, this paper provides a framework {of} conforming {mixed} finite element methods. Numerical experiments are given to validate the theoretical results.

Mixed Variational Formulation of Coupled Plates

Abstract

This paper proposes a mixed variational formulation for the problem of two coupled plates with a rigid {junction}. The proposed mixed {formulation} introduces {the union of} stresses and moments as {an auxiliary variable}, which {are} commonly of great interest in practical applications. The primary challenge lies in determining a suitable {space involving} both boundary and junction conditions of the auxiliary variable. The {theory} of densely defined operators in Hilbert spaces is employed to define {a nonstandard Sobolev space} without the use of trace operators. The well-posedness is established for the mixed formulation. Based on these conditions, this paper provides a framework {of} conforming {mixed} finite element methods. Numerical experiments are given to validate the theoretical results.
Paper Structure (12 sections, 7 theorems, 68 equations, 6 figures, 6 tables)

This paper contains 12 sections, 7 theorems, 68 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypotheses and notations
  4. Two coupled plates model
  5. Variational formulation based on displacements
  6. Mixed Variational Formulation
  7. Spaces and operators
  8. Mixed formulation and its well-posedness
  9. Continuity conditions
  10. Mixed finite element methods
  11. Mixed finite element methods
  12. Numerical test

Key Result

Lemma 2.1

The bilinear form $D(\bullet, \bullet)$ defined in D is continuous and coercive on $W \times W$. Moreover, the solution $\phi$ depends continuously on $F$ with a constant $c$, namely,

Figures (6)

  • Figure 2.1: (A) illustrates the shape and force distribution under the underformed state; (B) and (C) depict the deformed shapes when subjected solely to in-plane and out-of-plane loads, respectively.
  • Figure 2.2: The local coordinate systems of two coupled plates.
  • Figure 4.1: Left and Right are degrees of freedom for a $P_3$-$H(\operatorname{div}, \mathbb{S})$ element and a discontinous vectorial $P_2$ element, respectively.
  • Figure 4.2: Left and Right are degrees of freedom for a $P_4$-$H(\operatorname{divDiv},\mathbb{S})$ element and a discontinous $P_2$ element, respectively.
  • Figure 4.3: The coupled plates with the rigid junction of Example \ref{['ex2']}.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 2.1: Well-posednesslie1992mathematical
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • ...and 4 more