Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Energy Equilibrium and Newton CG Method

In course

CS-457: Geometric computing

This course will cover mathematical concepts and efficient numerical methods for geometric computing. We will explore the beauty of geometry and develop algorithms to simulate and optimize 2D and 3D geometric models with an emphasis towards computational d ...

Description

This lecture covers topics related to continuum mechanics, including reference and deformed configurations, deformation gradient, energy density function, elastic energy, strain, stress, linear elasticity, hyperelastic materials, St. Venant-Kirchhoff model, Corotated linear elasticity, Neo-Hookean model, force balance in 3D solids, Lamé parameters, equilibrium conditions, isotropic linear elasticity, surface traction, force balance, divergence of a vector field, divergence theorem, finite element discretization, energy minimization, Newton's method, conjugate gradients, iterative linear systems solver, gradient descent, conjugacy, conjugate direction method, conjugate gradient method, line search Newton-CG method, and outlook on implementing Newton CG for elastic equilibrium solve.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Mark Pauly

Mark Pauly is a full professor at the School of Computer and Communication Sciences at EPFL. Prior to joining EPFL, he was assistant professor at the CS department of ETH Zurich since April 2005. From August 2003 to March 2005 he was a postdoctoral scholar at Stanford University, where he also held a position as visiting assistant professor during the summer of 2005. He received his Ph.D. degree (with distinction) in 2003 from ETH Zurich and his M.S. degree (with highest honors) in 1999 from TU Kaiserslautern. His research interests include computer graphics and animation, shape modeling and analysis, geometry processing, architectural geometry, and digital fabrication. He received the ETH medal for outstanding dissertation, was awarded the Eurographics Young Researcher Award in 2006 and the Eurographics Outstanding Technical Contributions Award in 2016.

Official source

Ontological neighbourhood

Chemical engineering

Materials science: Deformation of materials

Related concepts (32)

Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.

Finite strain theory

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Gradient descent

In mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.

Deformation (physics)

In physics and continuum mechanics, deformation is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. Strain is related to deformation in terms of relative displacement of particles in the body that excludes rigid-body motions.

Related lectures (43)

Deformation and Strain Tensors

Explores deformation and strain tensors, Lagrange representation, elasticity theory, and the divergence theorem.

Theory of Elasticity

Explores the theory of elasticity, including Hooke's law, strain tensors, stress tensor, and thermodynamics of deformation.

Finite Element Analysis: Advanced Mechanisms in Engineering

Provides an overview of advanced mechanisms analysis using Finite Element Method and Finite Element Analysis in engineering applications.

Mechanics of Continuous Media

Explores stress-strain relationships, equilibrium of forces, and material properties in mechanics.

Solid mechanics in biomechanics: Large strain and stress definitions