Nonlinear Curved Simplicial Meshing with Guarantees

带保证的非线性曲线单纯网格划分

• 批准号: 451286978
• 负责人: Professor Dr. Marcel Campen
• 金额: --
• 依托单位: Institut für Informatik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/451286978?language=en
• 关键词: Nonlinear; Curved; Simplicial; Meshing; Guarantees

Partitioning complex 2D or 3D objects into structurally simple elements (such as triangles or tetrahedra) is at the heart of computational tasks in simulation, analysis, design, fabrication, animation, and computer graphics. Based on meshes of such elements, function spaces can be defined that these computational tasks can rigorously build on. The demand for such meshes has led to major research efforts in the field of algorithmic mesh generation over the past decades. The general task is: given a description of a 2D or 3D object’s boundary, automatically construct a mesh representation of its interior, meeting application-dependent quality requirements.The main focus has been on linear meshes, with straight edges. In the common case that an object’s boundary is not piecewise planar, however, they can only approximately represent the object; the mesh is not conforming with the boundary. Accurate conformance, however, is an important ingredient in demanding applications, e.g., in numerical analysis, physical simulation, relevant for accuracy and efficiency. An exact, error-free representation is enabled by the use of more general nonlinear elements. In particular, higher-order polynomial or rational curved elements are able to exactly conform to industry standard curved object boundary representations. The potential of methods built on such higher-order meshes has been demonstrated manifoldly.An ideal method for the generation of such nonlinear meshes can be expected to yield elements that are 1) boundary-conforming and 2) regular. A higher-order element is regular if it is defined through a polynomial or rational map of that order that is injective – an important prerequisite, e.g., in the finite element method and related techniques. Common higher-order meshing approaches, however, reliably achieve only one of these two important properties in general, not both. A recent result from the PI’s research group is a novel strategy that explicitly and systematically guarantees both properties by construction. It concerns nonlinear triangle meshes for 2D domains with curved boundary, and can be viewed as this project’s point of departure.In this project reliable algorithms for the generation of valid meshes of provably regular and conforming nonlinear elements are targeted. While the preliminary result is restricted to the special case of 2D piecewise polynomial boundaries, the goal is to support the general regime of practically relevant cases: 2D and 3D domains, polynomial and rational boundaries, C0 and higher-order continuity. This will fill a gap in the set of techniques required to further advance the utility and applicability of higher-order mesh based methods. It can relieve applications from the robustness issues still prevalent in the nonlinear mesh generation stage today – which are particularly pressing in increasingly common scenarios that require a fully automatic handling of large collections of objects or object variations.

将复杂的2D或3D对象划分为结构简单的元素（如三角形或四面体）是模拟、分析、设计、制造、动画和计算机图形学中计算任务的核心。基于这些元素的网格，函数空间可以被定义为这些计算任务可以严格地建立在。对这种网格的需求导致了在过去几十年中在算法网格生成领域的主要研究工作。总的任务是：给定2D或3D对象边界的描述，自动构建其内部的网格表示，满足应用程序相关的质量要求。主要关注具有直边的线性网格。然而，在对象的边界不是分段平面的常见情况下，它们只能近似地表示对象;网格与边界不一致。然而，准确的一致性是要求苛刻的应用中的重要因素，例如，在数值分析、物理模拟中，关系到精度和效率。通过使用更一般的非线性元件，能够实现精确的、无误差的表示。特别是，高阶多项式或有理曲线元素能够完全符合行业标准的弯曲对象边界表示。建立在这种高阶网格上的方法的潜力已经被证明是多样的。一个理想的方法来生成这样的非线性网格可以预期产生的元素是1）边界协调和2）规则。高阶元素是正则的，如果它是通过该阶的多项式或有理映射定义的，该阶是单射的-这是一个重要的先决条件，例如，有限元方法及相关技术。然而，常见的高阶网格划分方法通常只能可靠地实现这两个重要特性中的一个，而不能同时实现这两个特性。PI研究小组最近的一项成果是一种新的策略，通过构建明确而系统地保证了这两种特性。它是关于二维曲面区域的非线性三角形网格的，可以看作是本项目的出发点，本项目的目标是生成可证明为正则和协调的非线性单元的有效网格的可靠算法。虽然初步结果仅限于二维分段多项式边界的特殊情况下，我们的目标是支持一般制度的实际相关的情况下：二维和三维域，多项式和合理的边界，C 0和高阶连续性。这将填补一个空白，在一套技术需要进一步推进的效用和适用性的高阶网格为基础的方法。它可以减轻应用程序的鲁棒性问题仍然普遍存在的非线性网格生成阶段的今天-这是特别紧迫的日益普遍的情况下，需要全自动处理大量的对象或对象的变化。