By Dr. Barna Szabó
Engineering Software Research and Development, Inc.
St. Louis, Missouri USA

Meshless methods, also known as mesh-free methods, are computational techniques used for the approximation of the solutions of partial differential equations in the engineering and applied sciences. The advertised advantage of the method is that users do not have to worry about meshing. However, eliminating the meshing problem has introduced other, more complex issues. Oftentimes, advocates of meshless methods fail to mention their numerous disadvantages.

When meshless methods were first proposed as an alternative to the finite element method, creating a finite element mesh was more burdensome than it is today. Undoubtedly, mesh generation will become even less problematic with the application of artificial intelligence tools, and the main argument for using meshless methods will weaken over time.

An artistic rendering of the idea of meshless clouds. The spheres represent the supports of the basis functions associated with the centers of the spheres. Image generated by Microsoft Copilot.

Setting Criteria

First and foremost, numerical solution methods must be reliable. This is not just a desirable feature but an essential prerequisite for realizing the potential of numerical simulation and achieving success with initiatives such as digital transformation, digital twins, and explainable artificial intelligence, all of which rely on predictions based on numerical simulation. Assurance of reliability means that (a) the data and parameters fall within the domain of calibration of a validated model, and (b) the numerical solutions have been verified.

In the following, I compare the finite element and meshless methods from the point of view of reliability. The basis for comparison is the finite element method as it would be implemented today, not as it was implemented in legacy codes which are based on pre-1970s thinking. Currently, ESRD’s StressCheck is the only commercially available implementation that supports procedures for estimating and controlling model form and approximation errors in terms of the quantities of interest.

The Finite Element Method (FEM)

The finite element method (FEM) has a solid scientific foundation, developed post-1970. It is supported by theorems that establish conditions for its stability, consistency, and convergence rates. Algorithms exist for estimating the relative errors in approximations of quantities of interest, alongside procedures for controlling model form errors [1].

The Partition of Unity Finite Element Method (PUFEM)

The finite element method has been shown to work well for a wide range of problems, covering most engineering problems. However, it is not without limitations: For the convergence rates to be reasonable, the exact solution of the underlying problem has to have some regularity. Resorting to alternative techniques is warranted when standard implementations of the finite element method are not applicable.  One such technique is the Partition of Unity Finite Element Method (PUFEM), which can be understood as a generalization of the h, p, and hp versions of the finite element method [2]. It provides the ability to incorporate analytical information specific to the problem being solved in the finite element space.

The FEM Challenge

Any method proposed to rival FEM should, at the very least, demonstrate superior performance for a clearly defined set of problems. The benchmark for comparison should involve computing a specific quantity of interest and proving that the relative error is less than, for example, 1%. I am not aware of any publication on meshless methods that has tackled this challenge.

Meshless Methods

Various meshless methods, such as the Element-Free Galerkin (EFG) method, Moving Least Squares (MLS), and Smoothed Particle Hydrodynamics (SPH), using weak and strong formulations of the underlying partial differential equations, have been proposed. The theoretical foundations of meshless methods are not as well-developed as those of the Finite Element Method (FEM). The users of meshless methods have to cope with the following issues:

1. Enforcement of boundary conditions: The enforcement of essential boundary conditions in meshless methods is generally more complex and less intuitive than in FEM. The size of errors incurred from enforcing boundary conditions can be substantial.
2. Sensitivity to the choice of basis functions: The stability of meshless methods can be highly sensitive to the choice of basis functions.
3. Verification: Solution verification with meshless methods poses significant challenges.
4. Most meshless methods are not really meshless: It is true that traditional meshing is not required, but in weak formulations, the products of the derivatives of the basis functions have to be integrated. Numerical integration is performed over the domains defined by the intersection of supports (support is the subdomain on which the basis function is not zero), which requires a “background mesh.”
5. Computational power: Meshless methods often require greater computational power due to the global nature of the shape functions used, which can lead to denser matrices compared to FEM.

Advice to Management

Decision-makers need solid evidence supporting the reliability of data generated by numerical simulation. Otherwise, where would they get their courage to sign the “blueprint”? They should require estimates of the error of approximation for the quantities of interest. Without such estimates, the value of the computed information is greatly diminished because the unknown approximation errors increase uncertainties in the predicted data.

Management should treat claims of accuracy in marketing materials for legacy finite element software and any software implementing meshless methods with a healthy dose of skepticism. Assertions that a software product was tested against benchmarks and found to perform well should never be taken to mean that it will perform similarly well in all cases. Management should require problem-specific estimates of relative errors in the quantities of interest.

References

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