Mesh Generation: A Quick Introduction

MESHING
RESEARCH
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Meshing can be defined as the process of breaking up a physical domain into smaller sub-domains (elements) in order to facilitate the numerical solution of a partial differential equation. While meshing can be used for a wide variety of applications, the principal application of interest is the finite element method. Surface domains may be subdivided into triangle or quadrilateral shapes, while volumes may be subdivided primarily into tetrahedra or hexahedra shapes. The shape and distribution of the elements is ideally defined by automatic meshing algorithms.

The finite element method in recent decades has become a mainstay for industrial engineering design and analysis. Increasingly larger and more complex designs are being simulated using the finite element method. With its increasing popularity comes the incentive to improve automatic meshing algorithms.

At the inception of the finite element method, most users were satisfied to simulate vastly simplified forms of their final design utilizing only tens or hundreds of elements. Painstaking preprocessing was required to subdivide domains into usable elements. Market forces have now pushed meshing technology to a point where users now expect to mesh complex domains with thousands or millions of elements with no more interactions than the push of a button.

Consumers of finite element technology such as aerospace and automotive industries have immediate needs to shorten design cycles and overall time to market. Improving the robustness, speed and quality of automatic meshers, while only a small part of the entire process, can translate into increased revenue and competitive advantage.

While there is certainly the incentive from a market-based perspective to improve finite element meshing technology, opinions on the specifics of what should be improved are diverse. Amongst users of finite element technology their has long been a debate as to what shape of element produces the most accurate result. There is the often-held position that quadrilateral and hexahedral shaped elements have superior performance to triangle and tetrahedral shaped elements when comparing an equivalent number of degrees of freedom. Use of hex elements can also vastly reduce the number of elements and consequently analysis and post-processing times. In addition, hex and quadrilateral elements are more suited for non- linear analysis as well as situations where alignment of elements is important to the physics of the problem, such as in computational fluid dynamics or simulation of composite materials.

The automatic mesh generation problem is that of attempting to define a set of nodes and elements in order to best describe a geometric domain, subject to various element size and shape criteria. Geometry is most often composed of vertices, curves, surfaces and solids as described by a CAD or solids modeling package.

Many applications, use a "bottom-up" approach to mesh generation. Vertices are first meshed, followed by curves, then surfaces and finally solids. The input for the subsequent meshing operation is the result of the previous lower dimension meshing operation.

For example, nodes are first placed at all vertices of the geometry. Nodes are then distributed along geometric curves. The result of the curve meshing process provides input to a surface meshing algorithm, where a set of curves define a closed set of surface loops. Decomposing the surface into well-shaped triangles or quadrilaterals is the next phase of the meshing process. Finally, if a solid model is provided as the geometric domain, a set of meshed areas defining a closed volume is provided as input to a volume mesher for automatic formation of tetrahedra, hexahedra or mixed element types.

The journal articles referenced in this web site all are relevant to the process of mesh generation. Although many authors will take radically different approaches, the ultimate goal is to provide a mesh that can be used to solve a partial differential equation. This field is relatively new. Most of the papers published in this area have been within the past five to ten years. Many questions have been solved in this time, but there are still a significant number of problems to be addressed.