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What are the most effective techniques for generating meshes with good boundary approximation?

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Mesh generation is a fundamental task in many applications that involve geometric modeling, computer graphics, simulation, and analysis. A mesh is a collection of vertices, edges, and faces that represent the shape and topology of a complex object or domain. However, generating a mesh that accurately approximates the boundary of the original object or domain is not always easy, especially when the boundary is irregular, curved, or noisy. In this article, you will learn about some of the most effective techniques for generating meshes with good boundary approximation, and how they can improve the quality and performance of your mesh-based algorithms.

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1 Delaunay Triangulation

One of the most popular and widely used techniques for generating meshes is Delaunay triangulation. This technique creates a mesh of triangles that satisfy the Delaunay property, which means that no vertex of any triangle is inside the circumcircle of any other triangle. This property ensures that the triangles are as close to equilateral as possible, which minimizes the angles and maximizes the area of each triangle. Delaunay triangulation also has the advantage of being unique, optimal, and conforming to the boundary of the input points, as long as the points are in general position. However, Delaunay triangulation may not be able to handle some complex or degenerate cases, such as holes, sharp features, or non-manifold boundaries.

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Saeed Amiri

Senior Algorithm Engineer at Delivery Hero
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Triangles are amongst simplest objects we are getting familiar with since childhood. As a consequence triangulation is an extremely helpful tool for us to analyze complex spatial data. Delaunay triangulation meshes spatial data uniformly. Its applications includes area computation, shortest pathfinding with obstacles by exploiting triangulation graph, heatmaps, and robotics. Interestingly, voronoi diagrams and k-nearest neighbors, forming a dual relationship with Delaunay triangulation. If you are not familiar with voronoi diagrams, for illustration consider a bee hive; each cell's center forms a point for a Voronoi diagram. Connect neighboring centers, and we get the Delaunay Triangulation: showcasing the dual relationship between them.

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Girijesh Sharma

AI Research Scientist
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Delaunay triangulation is a widely used technique for generating triangular meshes with good geometric properties. It ensures that no vertex lies inside the circumcircle of any triangle in the mesh, resulting in well-shaped triangles.

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2 Advancing Front Method

Another technique for generating meshes is the advancing front method. This technique starts from a set of initial edges or faces that form the boundary of the object or domain, and then grows the mesh by adding new elements along the active edges or faces. The new elements are created by connecting the active edges or faces with nearby points or vertices that satisfy some criteria, such as distance, angle, or quality. The advancing front method can handle arbitrary boundaries, holes, and features, and can adapt the mesh size and shape to the local geometry and curvature. However, the advancing front method may require more user input and control, and may produce gaps, overlaps, or poor quality elements if the criteria are not well chosen.

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Girijesh Sharma

AI Research Scientist
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This method involves the propagation of a front through the domain, creating elements on-the-fly as it advances. It is particularly effective for structured or semi-structured mesh generation and can be adapted for unstructured meshing as well.

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3 Ruppert's Algorithm

A technique that combines the benefits of Delaunay triangulation and advancing front method is Ruppert's algorithm. This technique is also known as Delaunay refinement, and it aims to generate a mesh of triangles that are both Delaunay and well-shaped. Ruppert's algorithm starts from a Delaunay triangulation of the input points, and then iteratively improves the mesh by inserting new points at the circumcenters of bad triangles, or at the midpoints of long boundary edges. A bad triangle is a triangle that has an angle larger than a specified threshold, and a long boundary edge is an edge that is longer than a specified fraction of the local feature size. Ruppert's algorithm can produce meshes with good boundary approximation, smoothness, and quality, but it may also introduce many small triangles and increase the mesh complexity.

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Girijesh Sharma

AI Research Scientist
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Ruppert's algorithm, or Delaunay refinement, combines Delaunay triangulation and the advancing front method to iteratively enhance triangular mesh quality. It starts with a Delaunay triangulation, identifies "bad" triangles with large angles, and refines the mesh by inserting points at circumcenters. Long boundary edges trigger midpoint refinement. Iterative steps continue until predefined criteria are met. While effective for good boundary approximation, the trade-off is increased mesh complexity due to numerous small triangles. Despite this, Ruppert's algorithm is valuable for applications prioritizing high-quality meshes in finite element analysis and computational simulations.

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4 Quadtree-Based Method

A technique that uses a different approach for generating meshes is the quadtree-based method. This technique creates a mesh of quadrilaterals that are aligned with the coordinate axes, and that are derived from a hierarchical data structure called a quadtree. A quadtree is a tree that recursively subdivides a square region into four equal subregions, until some criterion is met, such as the number of points, the level of detail, or the boundary intersection. The quadtree-based method can generate meshes that are fast, simple, and easy to manipulate, and that can handle large and irregular domains. However, the quadtree-based method may not be able to capture the boundary shape and curvature accurately, and may produce jagged or distorted elements near the boundary.

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Girijesh Sharma

AI Research Scientist
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The quadtree-based method employs a distinct approach to mesh generation, constructing quadrilateral meshes aligned with coordinate axes through a hierarchical quadtree structure. The quadtree recursively subdivides a square region into four equal parts based on criteria like point density or boundary intersection. This method is known for its speed, simplicity, and ease of manipulation, making it suitable for large and irregular domains. However, limitations include potential inaccuracies in capturing boundary shape and curvature, leading to the creation of jagged or distorted elements near the boundary.

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5 Marching Cubes Algorithm

A technique that is specifically designed for generating meshes from volumetric data, such as images, scans, or simulations, is the marching cubes algorithm. This technique creates a mesh of triangles that approximate the iso-surface of a scalar field, which is a surface that has a constant value in the field. The marching cubes algorithm works by dividing the volume into a grid of cubes, and then determining the intersection of the iso-surface with each cube. The intersection is classified into one of 15 possible cases, based on the sign of the field value at the eight vertices of the cube. The cases are then mapped to a set of triangles that form the mesh. The marching cubes algorithm can generate meshes that are robust, efficient, and consistent, and that can represent complex and smooth surfaces. However, the marching cubes algorithm may also produce cracks, holes, or ambiguities in the mesh, depending on the resolution and quality of the data.

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Girijesh Sharma

AI Research Scientist
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The marching cubes algorithm is tailored for mesh generation from volumetric data, creating triangles approximating iso-surfaces in scalar fields. By dividing the volume into a grid of cubes, it identifies iso-surface intersections in each cube, classifying them into 15 cases based on vertex field values. These cases are then mapped to triangles, forming a mesh. The algorithm excels in generating robust, efficient, and consistent meshes that represent complex and smooth surfaces. However, challenges like cracks, holes, or ambiguities may arise, influenced by data resolution and quality.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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John Jurewicz

Senior Associate Architect
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Issues to Consider when establishing mesh networks for construction sites: Reflections, sensors that go bad and verifying that i-beacons and triangulation is all working, These are all considerations we have recently experienced with establishing mesh networks.

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Girijesh Sharma

AI Research Scientist
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Isogeometric Analysis (IGA): IGA employs the same basis functions for both geometry and simulation, providing a seamless integration of CAD and finite element analysis. This approach can result in high-quality meshes that closely approximate the geometry.

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What are the most effective techniques for generating meshes with good boundary approximation?

1

2

3

4

5

6

1 Delaunay Triangulation

2 Advancing Front Method

3 Ruppert's Algorithm

4 Quadtree-Based Method

5 Marching Cubes Algorithm

6 Here’s what else to consider

Algorithms

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What are the most effective techniques for generating meshes with good boundary approximation?

1

2

3

4

5

6

1 Delaunay Triangulation

2 Advancing Front Method

3 Ruppert's Algorithm

4 Quadtree-Based Method

5 Marching Cubes Algorithm

6 Here’s what else to consider

Algorithms

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