Volumetric parameterization and trivariate b-spline fitting using harmonic functions

Abstract

We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes T.J. 2005]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to [Martin and Cohen 2001] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis.

Biharmonic Volumetric Mapping Using Fundamental Solutions

“…In recent years, volumetric mapping have gained great interest due to its rich applications in many fields such as computer-aided manufacturing [8], meshing [9], [10], shape registration [11], [12], [13], and trivariate spline construction [14], [15], [16]. Wang et al [12] discretize the volumetric harmonic energy on the tetrahedral mesh using the finite element method, parameterized volumetric shapes over solid spheres by a variational algorithm.…”

“…Joshi et al [20] present harmonic coordinates with nonnegative weights for volumetric interpolation and deformation in concave regions. Martin et al [14] parameterize volumetric model with trivial topology to a cylinder using the finite element method, and later generalize the algorithm [15] to more complicated models with medial surfaces. Lipman et al [21] develop Green's coordinates for volumetric deformation.…”

Planar Parametrization in Isogeometric Analysis

“…As before, in the lower bound (31b) for det J, the number δ ∈ [0, 1] is an algorithmic parameter and the number Z is the result of the optimization (20). It goes without saying that if the Poisson problem is replaced by another problem, only the equations (31d)-(31f) are changed.…”

“…In the lower bound (22b) for det J, the number δ ∈ [0, 1] is an algorithmic parameter and the number Z is the result of the optimization (20). We have often successfully used δ = 0.…”

“…Many existing methods rely on the theory of harmonic functions on the physical domain. The method in [20] works on a triangulated volume and starts by constructing a parametrization of the boundary, i.e., the outer surface, using two harmonic functions with near orthogonal gradients. Then using harmonic functions in 3D the parametrization is propagated inwards to fill the entire volume.…”