Past 2013: Refer to SDGFEM YouTube channel (link)

Spacetime Discontinuous Galerkin Finite Element Method


The movies show the scattering of a planar wave from a stationary crack tip for linear elastodynamics. Due to symmetries only a quadrant of the mid-crack domain is analyzed. The three movies are:

  1. High gradient Visualization:  The gradient magnitude of finite element solutions are mapped to color. The sharp planar wave front approaching the crack tip, scatterings off of crack tip, and Raleigh and diagonal waves can be observed.
  2. Spacetime meshing animation: In the SDG method, the solution advances in time (up direction) by solving a small patch of elements at a time. This yields the method’s linear solution complexity and lends itself to adaptive and parallel computing. The highly refined regions correspond to moving wave fronts. The spacetime mesh contains 11 million tetrahedra and the refinement ratio for this h-adaptive simulation is smaller than 10^{-4}.
  3. Solution visualization: The velocity magnitude and log of strain energy are mapped to height and color fields respectively.

Contact mechanics using our dynamically consistent Riemann solutions

Our Riemann solutions for dynamic contact preserve the characteristic structure of the underlying elastodynamics problem and almost entirely eliminate common numerical artifacts observed at various contact mode transitions. These solutions also exhibit that the stick-slip transition is actually continuous for isotropic contact and no numerical regularization is needed. We have devised a regularization with maximum allowable penetration for separation to contact transition, the only one that physically generates solution discontinuities. The movies are:

  1. Brake pad sliding: This simulation on simplified geometry demonstrates the development of stick-slip-separation mode transitions and contact instabilities. For this high sliding velocity, separation waves start to propagate from the left to the right.
  2. Frictional contact of an elastic square against a rigid surface. The apparent penetration is caused by the use of perspective view.

Height field: velocity magnitude; Color field: strain energy.

An integrated interfacial contact/fracture model

A rate-dependent interfacial damage model represents the process of void nucleation, growth, and coalescence on the fracture surface. The contact modes are seamlessly integrated into the damage model. The movies are:

  1. Mid-crack contact/fracture under cyclic load.
  2. Inclusion-matrix debonding under cyclic load.

Height field: velocity magnitude; Color field: log of strain energy.

Quasi-singular velocity response and dynamic fracture using cohesive models

The Linear Elastic Fracture Mechanics (LEFM) theory predict a 1/√r singular response for the stress and velocity fields around a moving crack tip. Although cohesive models remove the stress singularity, we showed that the velocity field exhibits a quasi-singular behavior in that the material velocities tend to infinity as the crack speed approach the Raleigh wave speed.

  1. Quasi-singular velocity response: As the crack accelerates, the material velocity (height field) around the crack tip increases.
  2. zoom out view of 1.
  3. Inclusion-matrix debonding using a cohesive traction separation relation under tensile load.

Height field: velocity magnitude; Color field: log of strain energy.

Dynamic brittle fracture using a probabilistic nucleation model

The SDG’s powerful adaptive operations align element boundaries with a given crack direction in spacetime. While cracks can propagate in any directions, unlike many existing methods no inter-element discontinuity functions are needed. The use of a novel probabilistic crack nucleation approach is directly responsible for all brittle fracture features; no additional criteria for microcracking and crack branching are used. The movies are:

  1. Spatial discretization: The propagation of fracture surfaces are displayed on the spatial mesh. The areas around the crack tips are highly refined (e.g. refinement ratios smaller than 10^{-6}) and they are coarsened after further crack propagation and complete debonding.
  2. Deformed geometry visualization for crack propagation.
2 1/2 dimensional plate model and structural health monitoring

A 2 1/2 plate implementation combines the simplicity of a 2D discretization and the accuracy and flexibility of a 3D model. For Structural Health Monitoring (SHM) applications, the need to resolve minute differences caused by defects at their onset and distinguish them from numerical artifacts and to solve many forward problems for locating them require very accurate, stable, and efficient numerical methods. Local solution feature, linear solution complexity, and attractive adaptive and multiscale features of the SDG method makes it an ideal platform for SHM simulations. The movies are:

  1. Multiscale arrays of defects: Detection of defects in a plate from the scattering events off of arrays of defects (large defects at lower right corner and very small ones at upper left corner that cannot be seen). The highly multiscale nature of this problem can pose serious challenges in its numerical modeling.
  2. SHM for a composite laminate: The study of wave scattering off of a hole from a sinusoidal impulse load in a 2-ply composite plate.

Please also visit youtube SDG FEM channel for up-to-date animations.