PDE blending surfaces with C continuity

In this paper, we propose to use a general sixth-order partial differential equation (PDE) to solve the problem of C² continuous surface blending. Good accuracy and high efficiency are obtained by constructing a compound solution function, which is able to both satisfy the boundary conditions exactly and minimise the error of the PDE. This method can cope with much more complex surface-blending problems than other published analytical PDE methods. Comparison with the existing methods indicates that our method is capable of generating blending surfaces almost as fast and accurately as the closed-form method and it is more efficient and accurate than other extant PDE-based methods.     

Generating Blending Surfaces with a Pseudo-Lévy Series Solution to Fourth Order Partial Differential Equations

In our previous work, a more general fourth order partial differential equation (PDE) with three vector-valued parameters was introduced. This equation is able to generate a superset of the blending surfaces of those produced by other existing fourth order PDEs found in the literature. Since it is usually more difficult to solve PDEs analytically than numerically, many references are only concerned with numerical solutions, which unfortunately are often inefficient. In this paper, we have developed a fast and accurate resolution method, the pseudo-Lévy series method. Due to its analytical nature, the comparison with other existing methods indicates that the developed method can generate blending surfaces almost as quickly and accurately as the closed form resolution method, and has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods. In addition, it can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method. To demonstrate the potential of this new method we have applied it to various surface blending problems, including the generation of the blending surface between parametric primary surfaces, general second and higher degree surfaces, and surfaces defined by explicit equations.     

Analytical C smooth blending surfaces

Two factors are important in the generation of blending surfaces for interactive graphical and CAD applications, computational speed and the degree of smoothness. Most surface-blending methods blend surfaces with tangent continuity. However, curvature continuity has recently become increasingly important in various applications. In this paper, we present a method that is able to achieve curvature continuity based on the use of partial differential equations (PDE). The blending surfaces are generated as the solution to a sixth-order PDE with one vector-valued parameter. To achieve interactive performance, we propose an effective analytical method for the resolution of this sixth-order PDE.     

Controllable C1 continuous blending of time-dependent parametric surfaces

This paper proposes the concept of blending time-dependent varying surfaces, and develops a new method to create a controllable C1 continuous blending surface between primary parametric surfaces whose position and shape change with time. We treat it as a boundary-valued problem defined by the mathematical model of a vectored dynamic fourth-order partial differential equation subjected to time-dependent C1 continuous blending boundary constraints. High performance blending surface generation is achieved through the development of an approximate analytical solution of the mathematical model. We investigate the accuracy and efficiency of the solution, study the effective shape control of the blending surfaces, and apply the obtained solution to tackle surface blending problems. The applications demonstrate that our proposed approach is very effective and efficient in dealing with controllable C1 continuous surface blending between time-dependent varying parametric surfaces.     

基于Helmholtz 方程的过渡曲面设计

阐述了二阶和四阶Helmholtz 方程的一类周期边界问题的差分解法及其在 过渡曲面设计中的应用。这类方法不同于传统的PDE 方法中的二阶和四阶的偏微分方程, 比传统的二阶和四阶偏微分方程有了更多的自由项,因此,在曲面设计的时候,就有更多的 形状控制参数可进行调整,文中重点讨论了方程中的系数对曲面形状的影响,并研究了边界 切矢条件对曲面形状的影响及其在曲面形状设计中的应用。设计者只需给出边界曲线和边界 切矢,并通过对它们的控制就可构造和修改曲面形状。     

Higher-order nonlinear priors for surface reconstruction

For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.     

基于物理的能量曲面造型方法构造过渡曲面

讨论了过渡曲面的生成问题,指出了半径过渡和PDE方法构造过渡面的局限性,提出了用基于物理的能量曲面造型方法构造过渡曲面的方法,特别是用于解决管状封闭非周期性曲面和其他曲面间的过渡问题。该文详细讨论了基于物理的能量曲面造型方法构造过渡曲面的原理及求解方法,并给出了飞行器的翼身过渡和三通过渡的实例。     

Parameterization and reconstruction from 3D scattered points basedon neural network and PDE techniques

Reverse engineering ordinarily uses laser scanners since they can sample 3D data quickly and accurately relative to other systems. These laser scanner systems, however, yield an enormous amount of irregular and scattered digitized point data that requires intensive reconstruction processing. Reconstruction of freeform objects consists of two main stages: parameterization and surface fitting. Selection of an appropriate parameterization is essential for topology reconstruction as well as surface fitness. Current parameterization methods have topological problems that lead to undesired surface fitting results, such as noisy self-intersecting surfaces. Such problems are particularly common with concave shapes whose parametric grid is self-intersecting, resulting in a fitted surface that considerably twists and changes its original shape. In such cases, other parameterization approaches should be used in order to guarantee non-self-intersecting behavior. The parameterization method described in this paper is based on two stages: 2D initial parameterization; and 3D adaptive parameterization. Two methods were developed for the first stage: partial differential equation (PDE) parameterization and neural network self organizing maps (SOM) parameterization. The Gradient Descent Algorithm (GDA) and Random Surface Error Correction (RSEC), both of which are iterative surface fitting methods, were developed and implemented     

基于动态偏微分方程构造过渡面

为了实现交互式的偏微分方程曲面造型,针对传统静态偏微分方程构造过渡面存在的不足,提出了基于动态偏微分方程构造C1连续的过渡面,并引入迭代有限差分法求解偏微分方程的数值解,在此基础上构造了光滑过渡曲面.讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,其中形状控制因子对生成曲面的形状影响最为明显.造型实例表明,利用动态偏微分方程构造过渡面具有更高的灵活性,大大提高了工业几何设计的交互性,在CAD/CAM中具有重要的应用价值.     

Closed-form approximation of the periodic response of a system with a saturating non-linearity†

A technique for approximating harmonic, periodic solutions of a class of non-linear equations involving u saturation non-linearity is applied to a fourth-order equation which describes an electrical circuit containing a non-linear amplifying element. The computational effort required to obtain the closed-form approximation which this method yields is compared with that required for a numerical solution by the conventional methods of quasilinearization and patching. The various methods use comparable amounts of computer time, but the other classical methods do not offer the advantage of the closed-form approximation which our technique provides.     

An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the \(L^2\)-norm and \(L^{\infty }\)-norm for the solution and its gradient when the exact solution belongs to \(C^6\). Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.     

An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

We improve upon a method introduced in Bertalmio et al. [4] for solving evolution PDEs on codimension-one surfaces in As in the original method, by representing the surface as a level set of a smooth function, we use only finite differences on a Cartesian mesh to solve an Eulerian representation of the surface PDE in a neighborhood of the surface. We modify the original method by changing the Eulerian representation to include effects due to surface curvature. This modified PDE has the very useful property that any solution which is initially constant perpendicular to the surface remains so at later times. The change remedies many of problems facing the original method, including a need to frequently extend data off of the surface, uncertain boundary conditions, and terribly degenerate parabolic PDEs. We present numerical examples that include convergence tests in neighborhoods of the surface that shrink with the grid size Work supported by the National Science Foundation     

Direct raytracing of a closed-form fluid meniscus

We present a direct raytracing method for implicitly described fluid surfaces that takes into account the effects of capillary solid coupling at the boundaries. The method is independent of the underlying fluid simulation method and solely based on distance fields. We make use of the closed-form solution of the meniscus shape at the fluid interface to achieve the effect of surface tension exerted by the solid object. The shape of the liquid at these boundaries is influenced by various physical properties such as the force of gravity and the affinity between the liquid and the solid material. We generate contact angles at the boundaries without the need for computationally intensive small-scale simulation. At render time, we combine the closed-form solution for a small-scale effect with the numerical solution of a large-scale simulation. Our method is applicable for any implicit representation of the fluid surface and does not require an explicit extraction of the surface geometry. Therefore, it is especially useful for particle-based simulations. Furthermore, the solution is guaranteed to yield the correct contact angle and, for certain scenarios, it delivers the entirely correct solution throughout the interface; even in general scenarios, it yields plausible results. As for an example, we implemented and tested the proposed method in the setting of a smoothed particle hydrodynamics (SPH) fluid simulation.     

基于结式的曲面拼接与三维造型

针对很多几何造型是带有约束条件的曲面拼接问题,在线性连续同伦的基础上提出了利用非线性同伦连续计算拼接曲面以进行三维造型的方法。首先,根据得到的截面(切片)的位置及其曲线方程确定插值点并得到插值多项式;其次,将此插值多项式作为非线性连续同伦映射函数并分别代入主曲面和辅助曲面的多项式方程得到过渡曲面的方程;然后,仅将插值变元作为变元而主、辅助曲面方程的变元作为参数,利用Sylvester结式消去过渡方程中的变元得到关于主曲面的拼接方程即造型曲面。利用该方法能实现带有控制点的曲面造型以及多曲面约束的几何造型,而且它可以确定造型过程中的中间形状及中间形状的位置,从而更加具有实用性。     

PDE模型在声纳图像去噪中的应用研究

偏微分方程方法在光学图像去噪中已有很多成功的应用,但用于声纳图像去噪的情况还不多见。针对声纳图像受噪声污染严重的问题,将偏微分方程原理引入到声纳图像去噪中,重点讨论了两种偏微分方程模型:ROF模型和四阶扩散模型。基于这两种模型对声纳图像进行去噪处理,仿真实验证明了偏微分方程去噪算法的有效性,并对比分析了两种模型的去噪性能。ROF模型适用于低信噪比条件下的声纳图像处理,而四阶扩散模型在高信噪比条件下,能够很好地保持图像边缘,但当噪声污染严重时,其去噪后的SNR比ROF模型去噪低了近10 dB,不利于声纳图像处理。     

Discontinuous Galerkin Based Isogeometric Analysis for Geometric Flows and Applications in Geometric Modeling

We propose a method which combines isogeometric analysis with the discontinuous Galerkin (DG) method for second and fourth order geometric flows to generate fairing surfaces, which are composed of multiple patches. This technique can be used to tackle a challenging problem in geometric modeling–gluing multi-patches together smoothly to create complex models. Non-uniform rational B-splines (NURBS), the most popular representations of geometric models developed in Computer Aided Design, are employed to describe the geometry and represent the numerical solution. Since NURBS basis functions over two different patches are independent, DG methods can be appropriately applied to glue the multiple patches together to obtain smooth solutions. We present semi-discrete DG schemes to solve the problem, and \(\mathcal {L}^{2}\)-stability is proved for the proposed schemes. Our method enjoys the following advantages. Firstly, the geometric flexibility of NURBS basis functions, especially the use of multiple patches, enable us to construct surface models with complex geometry and topology. Secondly, the constructed geometry is fair. Thirdly, since only the control points of the NURBS patches evolve in accordance with the geometric flows, and their number (degrees of freedom) is very small, our algorithm is very efficient. Finally, this method can be easily formulated and implemented. We apply the method in mean curvature flows and in quasi surface diffusion flows to solve various geometric modeling problems, such as minimal surface generation, surface blending and hole filling, etc. Examples are provided to illustrate the effectiveness of our method.     

基于四阶偏微分方程的光滑曲面重构方法

常用的基于散点的曲面重构方法如克里金插值法、样条曲面拟合法等存在计算量大、重构曲面不光滑或无法插值已知散点等问题。为此,提出一种基于四阶偏微分方程的曲面重构方法。该方法首先选择一个四阶偏微分方程,并对其构建差分格式,进而分析该差分格式的稳定性和收敛性。在稳定性和收敛性条件下,采用演化的思想,通过有限差分法迭代求解偏微分方程的数值解,并将其稳态解作为原始曲面的逼近。以地质勘探中实际测井数据为例,采用偏微分方程曲面造型方法重构地质曲面,结果表明,该方法计算简便,构造的曲面具有自然光顺性且可以插值于已知散点。     

基于OpenGL的复杂曲面的纹理映射

介绍利用OpenGL和Visual C 6.0进行复杂曲面的纹理映射。利用解二次随圆偏微分方程的方法，得到任意曲面到平面的共形映射。此方法可以自动分配纹理坐标到复杂的没有起伏的曲面，有效地克服复杂曲面的自动纹理映射的变形，避免了纹理扰动。     

Efficient 2D Simulation on Moving 3D Surfaces

We present a method to simulate fluid flow on evolving surfaces, e.g., an oil film on a water surface. Given an animated surface (e.g., extracted from a particle-based fluid simulation) in three-dimensional space, we add a second simulation on this base animation. In general, we solve a partial differential equation (PDE) on a level set surface obtained from the animated input surface. The properties of the input surface are transferred to a sparse volume data structure that is then used for the simulation. We introduce one-way coupling strategies from input properties to our simulation and we add conservation of mass and momentum to existing methods that solve a PDE in a narrow-band using the Closest Point Method. In this way, we efficiently compute high-resolution 2D simulations on coarse input surfaces. Our approach helps visual effects creators easily integrate a workflow to simulate material flow on evolving surfaces into their existing production pipeline.