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This paper introduces a continuum mechanics based thread model for use in real-time simulation. The model includes both rotary inertia, shear deformation and torsion. It is based on a three-dimensional beam model, using a corotational approach for interactive simulation speeds as well as adaptive mesh resolution to maintain accuracy. Desirable aspects of this model from a numerical and implementation point of view include a true constant and symmetric mass matrix, a symmetric and easily evaluated tangent stiffness matrix, and easy implementation of time-stepping algorithms. From a modeling perspective interesting features are deformation of the thread cross section and the use of arbitrary cross sections without performance penalty.

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R-3. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in L-2 norm. This can be seen as an extension of the formalism and method originally used by Dziuk ( 1988) for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation inverted right perpendicular(h) of an implicitly defined surface inverted right perpendicular we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on inverted right perpendicular. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

In the mid '90s Delfour and Zolesio [4-6] established elasticity models on surfaces described using the signed distance function, an approach they called intrinsic modeling. For problems in codimension-two, e.g. one-dimensional geometries embedded in R³, an analogous description can be done using a vector distance function. In this paper we investigate the intrinsic approach for the modeling of codimension-two problems by deriving a weak formulation for a linear curved beam expressed in three dimensions from the equilibrium equations of linear elasticity. Based on this formulation we implement a finite element model using global degrees of freedom and discuss upon the effects of curvature and locking. Comparisons with classical solutions for both straight and curved cantilever beams under a tip load are given.