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Isogeometric analysis [1] has the advantage of integrating geometric and mesh models using NURBS or Spline. At the same time, I would like to ask a question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?

In detail, according to the discrete differential geometry, if we can find the equivalent definitions between continuous and discrete geometry, we can represent a continuous geometry with a mesh. We know that for CG software (for example Blender), the main way of modelling is to use the mesh to represent the continuous geometry, we modify the geometry by modifying the mesh. But for the scientific computing mesh generation software (for example Gmsh), we usually need a geometric model first and then set some parameters to do the meshing. There is a big difference between them. I know CG mainly uses the surface mesh in 3D space. And scientific calculation uses mainly solid mesh. But I think that CG's way of using mesh to represent geometry is simpler and suitable for shape change (optimization). So I wonder why can't we use mesh to represent geometry like CG in scientific computation. The geometry is directly represented by a mesh, and then you can do the computation directly on it. I think for the pioneers of FEM and FVM, the idea of combining the geometry and mesh must have been considered before. But now almost all the simulation software don't combine the geometry and mesh model. So I wonder what is the main difficulty of not doing that. So I would like to ask the question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?

This question is inspired by the answers of my previous question: Can the mesh generation methods in FVM and FEM be totally based on the knowledge of the mesh generation theory in computer graphics?

bibliography

1. Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng. 194, No. 39-41, 4135-4195 (2005). ZBL1151.74419.

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    $\begingroup$ Your question makes no sense to me. Formally, there is no fundamental difference between what you are referring to as "mesh" and what you are referring to as "geometry." The sophistication of the mathematical model representing the geometry depends on the requirements of the application using the geometry. $\endgroup$ – Bill Greene Aug 2 at 16:21
  • $\begingroup$ @BillGreene Thanks for the comment. What I mean is that for scientific computation, why can't the geometry be represented by the mesh directly. For scientific computation, the processes of geometric modelling and mesh generation are two distinctive processes. Why can't we merge them into one? As you said, we can choose the model of representing the geometry according to the application. $\endgroup$ – Dong Ivan Aug 2 at 22:42
  • $\begingroup$ In geometric modeling too, you have to create a geometry and then a mesh. Consider a Bezier curve. You specify the control points and then discretize according to the requirements of the view that you want to present to the world. A curve discretized with two points will just be a straight line. You need a finer 1D mesh to represent the curve better. And so one as you go to higher dimensions. $\endgroup$ – Biswajit Banerjee Aug 2 at 23:27
  • $\begingroup$ @BiswajitBanerjee Thanks for the comment. As I said in the question, there is a theory called discrete differential geometry which is fundamental in digital geometry processing. And according to that, if we can find the equivalent definitions between continuous and discrete geometry, we can represent a continuous geometry with a mesh(which is composed of point, straight lines, faces and solids). $\endgroup$ – Dong Ivan Aug 2 at 23:32
  • $\begingroup$ I would suggest that you check Arnold's monograph about Finite Element Exterior Calculus. $\endgroup$ – nicoguaro Aug 3 at 15:02
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The underlying problem is that the mesh really isn't the geometry. You want to simulate a bridge? It has a certain geometry, which you can approximate using a mesh, but the mesh is not the exact geometry. The same is true for most other curved objects.

There are of course approaches to integrate the geometry into the finite element mesh. In particular, I'd like to point you to this preprint. The primary obstacle is historical: the traditional workflow is to describe the exact geometry using CAD software; this CAD geometry is fed to the mesh generator that creates a mesh; the mesh is then given to the finite element ("analysis") software, but it no longer has access to the CAD geometry. The preprint linked to has an extensive description of what one needs to do to route the CAD geometry all the way through to the analysis engine, and why that is useful.

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  • $\begingroup$ is it possible that IgA will be a new (official) module in deal.II 9.3.0? $\endgroup$ – YSVSDVXCVXCV Aug 14 at 5:17
  • $\begingroup$ deal.II has a different way of representing geometries than IgA (which typically uses NURBS not just for the geometry but also for shape functions). But everything we say in the preprint is also part of deal.II 9.2 and, for the most part, in 9.1 already. $\endgroup$ – Wolfgang Bangerth Aug 14 at 13:35
  • $\begingroup$ is it implemented in "The step-60 tutorial of deal.II" $\endgroup$ – YSVSDVXCVXCV Aug 16 at 9:51
  • $\begingroup$ @YSVSDVXCVXCV There are a number of tutorial programs in the 50s and 60s range that show how to deal with geometry. $\endgroup$ – Wolfgang Bangerth Aug 17 at 19:33
  • $\begingroup$ Dear Proferssor, Thanks for your reply! I will check it carefully! $\endgroup$ – YSVSDVXCVXCV Aug 18 at 6:35
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"So I wonder why can't we use mesh to represent geometry like CG in scientific computation. The geometry is directly represented by a mesh, and then you can do the computation directly on it." ...

This is exactly what we do using iso-parametric elements in FEM; we discretise the geometry and field variables using the same mesh. The concept of iso-parametric elements is not limited to the Lagrange family of elements or NURBS. In theory, one can use any appropriate polynomial space for discretising the geometry and field variables, for example, subdivision surfaces, T-Splines, Box-splines, Chebyshev polynomials etc.

In FVM also the same mesh is used for both geometry and field variables. But the concept of iso-parametric representation is not applicable to FVM, especially for cell-centred FVM approaches. Please correct me if I am wrong.

IGA sounds very promising at the first-look. It certainly has it's advantages when compared to the traditional FEM. IGA is already available in LS-DYNA. But, I think that IGA's disadvantages at the moment outweigh it's advantages when it comes to practical applications. There are also other issues that are not technical, see point 5.

1.) IGA is not free from the discretisation of the space. One still has to discretise the geometry (to differentiate it from the word "mesh", if that makes some sense). Talking in terms of IGA, one needs to generate a new knot-space for NURBS. Such a feature is not yet available in CAD software. It is mostly carried out either manually or using custom scripts in academic research groups.

2.) IGA is rooted in the idea of higher continuity across element boundaries. While higher continuities are advantageous in FEM for some specific problems especially for problems with smooth solutions, this point is not so advantageous for problems with discontinuities in the field variables. Moreover, higher continuities are limited to an individual patch without sharp corners.

Furthermore, discontinuities across elements are inherent to the cell-centred FVM. Therefore, I don't think that it makes sense to use IGA directly for FVM. (I don't have much experience in FVM to elaborate further on this).

3.) IGA is expensive when compared with the traditional FEM. Higher continuity means more non-zero entries in the (effective) stiffness matrix. While it is true that we can get accurate results using coarse meshes with higher-order elements, it doesn't pay off much to use higher than quadratic polynomials for practical applications. In my experience, quadratic polynomials are the optimum choice when taking accuracy and efficiency (runtime) into consideration.

4.) It is also difficult to extend IGA to advanced problems in solid mechanics, for example, plasticity and incompressible hyperelasticity, which require sophisticated formulations.

  • Projection-based methods, for example, F-bar (or B-bar as some call it) formulation, require the inversion of the patch-wise mass matrix. -> Expensive.
  • Taylor-Hood type elements (polynomial for pressure field is one order lower than that for displacement (or velocity) field) are not inf-sup stable. One needs to use sub-division stabilisation to ensure inf-sup stability. (similar to the Q1-iso-Q1 element). This has practical limitations.

It would have been fantastic if we didn't have to resort to the sophisticated formulations when using IGA. But, unfortunately, we still have to use those advanced formulations.

5.) Design and analysis workflows in the industry are already well-established based on the software tools that use traditional FEM and FVM. It is extremely difficult to convince industries to change their workflows to use new simulation paradigms unless we show them some quick rewards or significant savings.

It is safe to say that IGA is not matured enough to replace traditional FEM. It is hard to say when it will. But one can combine the concepts and tools from IGA and traditional FEM to improve upon the existing schemes, see paper1 and paper2.

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  • $\begingroup$ for 1) how about trimming technique? We can direct import CAD file for IgA in Ls Dyna ...... $\endgroup$ – YSVSDVXCVXCV Aug 11 at 16:59
  • $\begingroup$ Sure! We still need to discretise the domain to solve the PDE. $\endgroup$ – Chenna K Aug 12 at 13:08
  • $\begingroup$ how about adaptive strategy for the discretization problems in IgA? $\endgroup$ – YSVSDVXCVXCV Aug 16 at 9:54
  • $\begingroup$ The question is whether or not such strategies are possible but if such software tools are readily available to the design/analysis engineers. $\endgroup$ – Chenna K Aug 17 at 8:41
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Yes, you can use the same mesh to represent the geometry of your domain and to solve the PDE, that is something that you can do. For example, a square can be completely described by two triangles and you can use this mesh to solve your PDE. If the solution of your PDE is constant or linear this mesh would suffice but if you have higher gradients in your solution you would need more elements.

In general, a sequence of meshes is considered and there is a fine-enough mesh (for your application) that gives you an error under your desired tolerance. For example, solving

$$\nabla^2 u = -80 \sin(4x) \cos(8y)\, ,$$

for $(x, y) \in [0, \pi]^2$ requires a mesh that is a square. Nevertheless, using just two element would not give you a solution close to the analytic result $u = \sin(4x) \cos(8y)$.

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