Intro
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