Free-form Patches Research Articles

MODELING THE DYNAMICS OF DEFORMABLE OBJECTS BASED ON VOLUMETRIC PATCHES OF FREE FORMS

A method for solving nonlinear implicit numerical integration problems based on volumetric patches of free forms for modeling deformable objects with high-speed dynamics is proposed. This method improves the accuracy, consistency and controllability of deformation modeling and animation. The method is characterized by speed, accuracy, stability and uses optimization with region decomposition. The method is well suited for modeling deformable bodies with a large time step, in a wide range of deformation dynamics. We propose decomposed optimization, an optimization method based on free-form patches with region decomposition to minimize incremental potentials at each time step. The method uses quadratic matrix decomposition to combine non-overlapping subregions. The Hessian is evaluated once at the beginning of the time step. The advantages of the method are as follows. Geometric primitives and their mathematical models are proposed, which allow the reasonable application of these primitives to solve problems of volume-oriented modeling. Such requirements are met by volumetric patches of free forms based on analytical perturbation functions relative to the base triangles. The decomposed Hessian is constructed for each subregion and calculated using a set of vertices taken from a complete non-decomposed grid. The weights add the missing second-order Hessian data to the vertices of the subregions from the neighbors along the decomposition boundaries. Thanks to this, the descent is performed along the grid coordinates. There is no need to add gradients. During the descent, the gradient is determined. The Hessians under the region are calculated and factorized in parallel once per time step. They are used as an initializer at each iteration. Then the results are mixed together. This ensures stable and continuous high-quality modeling. An automated and reliable optimization method adapted for modeling nonlinear materials, high-speed dynamics and large deformations is proposed.

Method of anisotropic deformation of elastic materials based on free-form patches

A method of anisotropic deformation of elastic materials based on free-form patches for biomechanics problems is presented. The method provides an analysis of the anisotropic hyperelasticity of transverse isotropy. For this purpose, fast and efficient Newton solutions are used. An invariant of anisotropic deformation is proposed, with the help of which side states of energy are eliminated at large deformation. The method is also used to restore finite elements. Because of this, it is possible to model large deformations, including for grids containing degenerate elements.

Method of anisotropic deformation of elastic materials based on free-form patches

A method of anisotropic deformation of elastic materials based on free-form patches for biomechanics problems is presented. The method provides an analysis of the anisotropic hyperelasticity of transverse isotropy. For this purpose, fast and efficient Newton solutions are used. An invariant of anisotropic deformation is proposed, with the help of which side states of energy are eliminated at large deformation. The method is also used to restore finite elements. Because of this, it is possible to model large deformations, including for grids containing degenerate elements.

THE METHOD OF DEFORMATION OF THREE-DIMENSIONAL OBJECTS BASED ON FREE-FORM PATCHES

THE METHOD OF DEFORMATION OF THREE-DIMENSIONAL OBJECTS BASED ON FREE-FORM PATCHES

A Complementary Sensor Approach to Reverse Engineering

A complementary sensor technique for reverse engineering objects that are represented by a three-dimensional (3D) cloud data set is reported. The research focuses on objects whose surface form is manifest as a set of distinct free-form surface patches, each of which is enclosed by a boundary. The method incorporates three stages: (1) laser scanner-based digitization of all the free-form surface patches, (2) touch probe-based digitization of the surface patch boundaries, and (3) modeling of both data sets to create a complete B-spline curve and surface representation of the object. The patch boundary data, defined by the touch probe, is employed to segment the free-form surface data into the constituent patches. Furthermore, the boundary data is incorporated within a B-spline surface fitting process to constrain the boundaries. The two sensors functionally complement each other; the range sensor provides the required dense resolution of 3D points on the free-form surfaces whereas the touch probe accurately defines the patch boundaries. The method is ideal for objects comprised of both functional engineering features, e.g. bearing holes or precise mounting locators, and aesthetic features, such as hand grips or part covers. The touch probe is also ideal for digitizing boundaries where occlusion prevents the use of an optical digitizer. The laser-based sensor has an accuracy specification of 50 microns (over a 40-mm depth of field) whereas the touch probe is accurate to 4 microns over a 25-mm measurement length. An example part is modeled that has multiple free-form patches (defining the part’s outer cover) that require a large cloud data set for complete coverage. The corresponding patch boundaries accurately define the location of critical part mounting locations that require the touch probe’s precision.

有理境界Ｃ｀２´ Ｇｒｅｇｏｒｙパッチ

This paper introduces a new type of free-form patches named rational boundary C2 Gregory patch. It can be said an extension of C2 Gregory patch developed by Miura and Wang, which gives users the capability of designing curvature-continuous surfaces (G2 continous surfaces) with reasonable flexibilities, and also that of rational boundary Gregory patch proposed by Chiyokura et al., which is surrounded by rational Bézier curves and can be interpolated with the continuity of tangent planes (G1 continuity). As the name of the patch implies, its boundary consists of rational Bézier curves. Its derivation is explained and methods for G2 continuity are proposed to connect it with a rational Bézier patch and with another RBC2G patch. Finally, a G2 continuous interpolation method based upon such patches is discussed.

A new expression of Coons equation

A simple free-form surface equation based on Coons expression can be used to adjoin a ruled surface and a free-form surface with continuous cross-sectional derivatives guaranteed. Two adjacent free-form patches defined with this equation are C 1- continuous with respect to u and w.

Approximate conversion of rational splines

The exchange of data between different geometric modeling systems frequently frequires approximate conversions of rational Be´zier and B-spline curves and surfaces to lower degree integral B-spline representations. Similarly, processing of trimmed, free-form patches, such as their automated manufacture, may lead to high degree representations. Accurate conversion of such geometries to lower degree integral representations is required in many modeling environments. The objective of this paper is to present an algorithm to approximate non-uniform rational B-spline curves with non-uniform integral B-spline curves of lower degree. Global error bounds for the verification of the accuracy of the approximation are established. This algorithm forms the basis of a similar rational B-spline surface approximation procedure.