This paper provides a formal connexion between springs and continuum mechanics in the context of onedimensional
and two-dimensional elasticity. In a first stage, the equivalence between tensile springs and the finite element
discretization of stretching energy on planar curves is established. Furthermore, when considering a quadratic strain function of stretch, we introduce a new type of springs called tensile biquadratic springs. In a second stage, we extend this equivalence to non-linear membranes (St Venant-Kirchhoff materials) on triangular meshes leading to triangular biquadratic and quadratic springs. Those tensile and angular springs produce isotropic deformations parameterized by Young modulus and Poisson ratios on unstructured meshes in an efficient and simple way. For a specific choice of the Poisson ratio, 0.3, we show that regular spring-mass models may be used realistically to simulate a membrane behavior. Finally, the different spring formulations are tested in pure traction and cloth simulation experiments.
Month: February 2009
An Edge-Based Computationally Efficient Formulation of Saint-Venant Kirchhoff Tetrahedral Finite Elements
This article describes a computationally efficient formulation and an algorithm for tetrahedral finite-element simulation of elastic objects subject to Saint Venant-Kirchhoff (StVK) material law. The number of floating point operations required by the algorithm is in the range of 15% to 27% for computing the vertex forces from a given set of vertex positions, and 27% to 38% for the tangent stiffness matrix, in comparison to a well-optimized algorithm directly derived from the conventional Total Lagrangian formulation. In the new algorithm, the data is associated with edges and tetrahedron-sharing edge-pairs (TSEPs), as opposed to tetrahedra, to avoid redundant computation. Another characteristic of the presented formulation is that it reduces to that of a spring-network model by simply ignoring all the TSEPs. The technique is demonstrated through an interactive application involving haptic interaction, being combined with a linearized implicit integration technique employing a preconditioned conjugate gradient method.