Computing the polar decomposition with applications
SIAM Journal on Scientific and Statistical Computing
Fast polar decomposition of an arbitrary matrix
SIAM Journal on Scientific and Statistical Computing
Algorithms for the polar decomposition
SIAM Journal on Scientific and Statistical Computing
LAPACK: a portable linear algebra library for high-performance computers
Proceedings of the 1990 ACM/IEEE conference on Supercomputing
A parallel algorithm for computing the polar decomposition
Parallel Computing
Matrix computations (3rd ed.)
Modeling inelastic deformation: viscolelasticity, plasticity, fracture
SIGGRAPH '88 Proceedings of the 15th annual conference on Computer graphics and interactive techniques
Approximate simulation of elastic membranes by triangulated spring meshes
Journal of Graphics Tools
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
Topological features in 2D symmetric higher-order tensor fields
EuroVis'11 Proceedings of the 13th Eurographics / IEEE - VGTC conference on Visualization
Hi-index | 0.00 |
Visualization of fourth-order tensors from solid mechanics has not been explored in depth previously. Challenges include the large number of components (3x3x3x3 for 3D), loss of major symmetry and loss of positive definiteness (with possibly zero or negative eigenvalues). This paper presents a decomposition of fourth-order tensors that facilitates their visualization and understanding. Fourth-order tensors are used to represent a solid's stiffness. The stiffness tensor represents the relationship between increments of stress and increments of strain. Visualizing stiffness is important to understand the changing state of solids during plastification and failure. In this work, we present a method to reduce the number of stiffness components to second-order 3x3 tensors for visualization. The reduction is based on polar decomposition, followed by eigen-decomposition on the polar "stretch". If any resulting eigenvalue is significantly lower than the others, the material has softened in that eigen-direction. The associated second-order eigentensor represents the mode of stress (such as compression, tension, shear, or some combination of these) to which the material becomes vulnerable. Thus we can visualize the physical meaning of plastification with techniques for visualizing second-order symmetric tensors.