Discretization of NURBS Models for Meshless Isogeometric Analysis

Conceitos Básicos

The authors present an algorithm for generating nodes on NURBS surfaces to solve PDEs efficiently using meshless methods, bridging node generation with isogeometric analysis.

Resumo

The content discusses a new algorithm for generating nodes on NURBS surfaces to solve PDEs efficiently using meshless methods. It highlights the importance of quasi-uniform node sets and their quality for stable discretizations. The approach is demonstrated on various geometries, showcasing convergence rates and computational complexities.
The study focuses on Poisson's equation, linear elasticity (Navier-Cauchy equation), and transient heat transport, providing insights into the effectiveness of the proposed algorithm in solving these problems.
Key metrics such as minimal distance between nodes and fill distance within the domain are analyzed to assess node quality. The results demonstrate expected convergence rates and computational efficiency in solving complex PDEs on CAD geometries.

Estatísticas

Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary.
Node sets may be generated in several different ways, including iterative optimization, sphere-packing, QR factorization, and repulsion.
Quasi-uniformly-spaced node sets improve the stability of meshless methods.
Poisson disk sampling via advancing front methods constitutes the state of the art in generating quasi-uniformly spaced nodes.
The DIVG algorithm allows for variable spacing Poisson disk sampling on complicated domains in arbitrary dimension.
The sDIVG algorithm generalizes DIVG to parametric surfaces, focusing on sampling a single NURBS surface.

Citações

Principais Insights Extraídos De

Discretization of non-uniform rational B-spline (NURBS) models for meshless isogeometric analysis

by Urban Duh,Va... às arxiv.org 02-29-2024

https://arxiv.org/pdf/2303.02638.pdf

Perguntas Mais Profundas

How does the proposed NURBS-DIVG algorithm compare to traditional mesh generation techniques

The proposed NURBS-DIVG algorithm offers several advantages over traditional mesh generation techniques. One key difference is that NURBS-DIVG generates nodes directly on the boundaries represented by non-uniform rational B-splines (NURBS), allowing for more accurate representation of complex geometries. Traditional mesh generation methods, on the other hand, often involve partitioning the domain into a finite number of elements to cover it entirely. This can be time-consuming and may not always capture the intricacies of CAD models accurately.
Additionally, NURBS-DIVG provides a hierarchical approach to generating quasi-uniform node sets both on NURBS surfaces representing domain boundaries and within volumes enclosed by these surfaces. This results in improved stability and accuracy for meshless numerical discretizations compared to traditional mesh-based approaches.
In summary, the NURBS-DIVG algorithm offers a more efficient and accurate way to generate nodes for meshless analysis on CAD geometries compared to traditional mesh generation techniques.

What challenges might arise when applying meshless methods to CAD supplied geometries

Applying meshless methods to CAD supplied geometries can present several challenges. One significant challenge is ensuring that node distributions are uniform near boundaries while also being robust and automated for practical engineering applications. Current node generation approaches on CAD geometries may struggle with maintaining quasi-uniformity near boundaries or providing sufficient quality guarantees for stable numerical solutions.
Another challenge lies in determining whether a particular node lies inside or outside the model when discretizing interior volumes with boundary representations as parametric surfaces or collections of patches like those in CAD models. Ensuring accurate inside/outside tests can be crucial for generating reliable node sets within complex geometries.
Furthermore, dealing with sharp edges, concavities, or vertices in CAD models can pose challenges in achieving optimal spacing between nodes and maintaining overall distribution quality throughout the geometry.
Overall, addressing these challenges will be essential for effectively applying meshless methods to CAD supplied geometries.

How can advancements in node generation algorithms impact the broader field of computational mathematics

Advancements in node generation algorithms have far-reaching implications across computational mathematics fields. Improved algorithms for generating high-quality nodes play a critical role in enhancing the accuracy and efficiency of various numerical methods used in computational mathematics.
One major impact is seen in improving the stability and convergence properties of numerical solutions obtained through meshless methods such as radial basis function-generated finite differences (RBF-FD). By generating well-distributed nodes with appropriate spacing functions based on advanced algorithms like DIVG and sDIVG tailored specifically for complex domains represented by NURBS surfaces, researchers can achieve more reliable results when solving partial differential equations (PDEs) using these techniques.
Moreover, advancements in node generation algorithms contribute towards bridging gaps between different mathematical disciplines such as computational geometry, computer-aided design (CAD), finite element analysis (FEA), and isogeometric analysis (IGA). These developments enable seamless integration of cutting-edge technologies into diverse applications ranging from engineering simulations to scientific computing.