This volume is devoted to the study of hyperbolic free boundary problems possessing variational structure. Such problems can be used to model, among others, oscillatory motion of a droplet on a surface or bouncing of an elastic body against a rigid obstacle. In the case of the droplet, for example, the membrane surrounding the fluid in general forms a positive contact angle with the obstacle, and therefore the second derivative is only a measure at the contact free boundary set. We will show how to derive the mathematical problem for a few physical systems starting from the action functional, discuss the mathematical theory, and introduce methods for its numerical solution. The mathematical theory and numerical methods depart from the classical approaches in that they are based on semi-discretization in time, which facilitates the application of the modern theory of calculus of variations.
Author(s): Seiro Omata, Karel Svadlenka, Elliott Ginder
Series: SpringerBriefs in Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 98
City: Singapore
Preface
Contents
Symbols
1 Introduction
2 Physical Motivation
2.1 Membrane Collision with an Obstacle
2.2 Peeling of Adhesive Tape
2.2.1 Energy at Equilibrium
2.2.2 Time-Dependent Problem
2.2.3 Regularized Equation and Its Limit
2.3 Droplet Motion: A Volume-Preserving Problem
2.3.1 Volume-Constrained Problem Without Free Boundary
2.3.2 Volume-Constrained Problem with Free Boundary
2.3.3 A Coupled Model of Membrane and Fluid Motion
3 Discrete Morse Flow
3.1 Discrete Morse Flow for the Linear Wave Equation
3.2 Energy-Preserving Scheme for the Linear Wave Equation
3.3 Volume-Constrained Hyperbolic Problem
4 Discrete Morse Flow with Free Boundary
4.1 Weak Solution and Discrete Morse Flow
4.2 Construction of Approximate Solutions
4.2.1 Assumptions
4.2.2 Statement of Semi-discrete Problem
4.2.3 Minimizers Are Nonnegative
4.2.4 Existence of Minimizers
4.2.5 Energy Estimate
4.2.6 Subsolution Property
4.2.7 L∞-Boundedness
4.2.8 Hölder Continuity
4.2.9 Euler–Lagrange Equation
4.2.10 Approximate Weak Solution
4.3 Existence of Weak Solution in One Dimension
4.3.1 Convergent Subsequence
4.3.2 Limit Process
5 Energy-Preserving Discrete Morse Flow
5.1 Modified Discrete Morse Flow Scheme
5.2 Evolution of Energy
5.3 Approximate Weak Solution
5.4 Numerical Results
6 Numerical Examples and Applications
6.1 Finite Element Approximation
6.2 Examples of Applications
6.2.1 Volume-Preserving Hyperbolic Mean Curvature Flow
6.2.2 Elastodynamic Contact Problem
6.2.3 Elastic Shell Impact
References
