Mixed Variational Formulation of Two Coupled Plates with Rigid Junctions

Core Concepts

Proposing a mixed variational formulation for two coupled plates with a rigid junction.

Abstract

The content introduces a mixed variational formulation for two coupled plates with a rigid junction. It addresses challenges in determining suitable spaces for stresses and moments, employing densely defined operators in Hilbert spaces. Continuity conditions, framework for finite element methods, and numerical experiments are discussed.
Introduction

Elastic multi-structures in engineering.
Mathematical models for coupled plates.
Specific Multi-Structure

Focus on two coupled plates with a rigid junction.
Introduction of a mixed variational formulation.
Challenges and Solutions

Determining suitable spaces for stresses and moments.
Use of densely defined operators in Hilbert spaces.
Framework for Finite Element Methods

Conforming mixed finite element methods.
Choice of various finite elements.
Preliminaries

Model assumptions and notation conventions.
Deformation of two coupled plates.
Mixed Variational Formulation

Introduction of a mixed formulation.
Establishment of well-posedness.
Continuity Conditions

Specification of continuity conditions for smooth functions.
Extraction of essential conditions from Problem 1.

Stats

The proposed mixed formulation introduces the union of stresses and moments as an auxiliary variable.
The theory of densely defined operators in Hilbert spaces is employed.
Continuity conditions for stresses and moments are provided.

Quotes

"There are several reasons behind considering the new mixed formulation."

Key Insights Distilled From

Mixed Variational Formulation of Coupled Plates

by Jun Hu,Zhen ... at arxiv.org 03-28-2024

https://arxiv.org/pdf/2403.18217.pdf

Deeper Inquiries

How does the mixed variational formulation enhance practical applications

The mixed variational formulation enhances practical applications by allowing for the direct calculation of stresses and moments, which are crucial variables in structural analysis. By introducing the union of stresses and moments as an auxiliary variable, the formulation preserves the continuity requirements of these variables at junctions and boundaries. This is essential for accurately modeling the behavior of coupled plates with rigid junctions. Additionally, the mixed formulation provides a framework for conforming mixed finite element methods, which can be used to numerically solve the problem efficiently and accurately.

What are the limitations of using densely defined operators in nonstandard Sobolev spaces

One limitation of using densely defined operators in nonstandard Sobolev spaces is the complexity involved in defining suitable function spaces that incorporate boundary and junction conditions. While this approach allows for the establishment of well-posedness and continuity conditions, it can be challenging to determine the appropriate spaces for the auxiliary variables. Additionally, the use of densely defined operators may introduce technical complexities in the formulation and implementation of numerical methods, potentially leading to computational challenges.

How can the proposed framework of conforming mixed finite element methods be extended to other structural problems

The proposed framework of conforming mixed finite element methods can be extended to other structural problems by adapting the formulation and continuity conditions to suit the specific characteristics of the problem at hand. For different structural problems, such as beams, shells, or three-dimensional structures, the mixed variational formulation can be modified to incorporate the relevant variables and boundary conditions. By adjusting the finite element spaces and discretization techniques, the framework can be applied to a wide range of structural analysis problems, providing a versatile and robust approach to numerical simulations in structural engineering.