FEM Calculations: What to Pay Attention To?

In modern engineering, FEM Calculations: What to Pay Attention To? are crucial for precise structural analysis. The Finite Element Method (FEM) relies heavily on the quality of the mesh, which forms the backbone of the model by dividing the space into smaller, finite elements. This allows for the resolution of complex differential equations that describe stress, displacement, and other physical quantities in the object under study. Understanding advanced meshing techniques and selecting appropriate elements are essential for obtaining reliable results, especially in structures with complex geometry, composite materials, or dynamic analysis.

FEM Calculations: Types of Meshes in Analysis

In FEM calculations, there are several approaches to mesh creation, each varying in complexity and accuracy in representing geometry and analysis results. The constant element method is the most traditional approach, where the entire model geometry is divided into elements of uniform size and shape. This type of mesh is simple to implement and sufficient for simple structures with regular shapes. However, it may not accurately represent complex geometries, leading to errors in analysis.

A more advanced approach is the polynomial edge approximation method. In this technique, element edges are approximated using high-order polynomial functions, allowing for more precise representation of complex shapes without significantly increasing the number of mesh elements. Using polynomials up to the 4th or 6th order is relatively common, especially in analyses requiring high accuracy, such as dynamic analyses or modeling highly complex geometries. In some cases, particularly in advanced spectral analyses, higher-order polynomials, up to the 9th order, are used to achieve extremely high precision, though this technique requires greater computational resources and experience in its application.

Sources of Errors in FEM Calculations: Impact of Constant and Polynomial Elements

Errors in Finite Element Method (FEM) calculations stem from several key factors, the most significant being model discreteness, approximation of differential equations, and boundary conditions and loads. Dividing a continuous body into a finite number of elements inevitably introduces simplifications and inaccuracies. These errors can be particularly noticeable in areas with complex stress gradients, such as near singularities like sharp edges or vertices.

For constant elements (of uniform size and shape), computational errors may arise from insufficient mesh resolution, especially in areas with complex geometry or large stress gradients. If the mesh is too coarse, the model will not accurately represent the actual distribution of stresses and deformations, leading to underestimation or overestimation of results. Additionally, constant-sized elements often have limited ability to accurately represent curvilinear edges and surfaces, which can also increase computational error.

On the other hand, polynomial elements allow for better approximation of complex geometries by using higher-order polynomials to approximate element edges and surfaces. This enables more precise results with fewer elements, as these elements can more accurately represent changes in the stress field. However, errors associated with polynomial elements can occur if the polynomial degree is not appropriately matched to the model’s complexity. For example, too low a polynomial order may lead to inaccuracies similar to those in constant elements, while too high an order may lead to excessive computational complexity and convergence issues.

FEM Calculations: Types of Elements: 1D, 2D, 3D

In FEM calculations, selecting the appropriate type of elements for analysis depends on the characteristics of the structure under study and the purpose of the analysis itself. In FEM, we distinguish three basic types of elements: 1D, 2D, and 3D elements, each with its specific applications and advantages.

• 1D Elements: These elements represent linear structures such as rods, beams, or cables. They are used in analyses of slender structures where one dimension is dominant, and deformations and stresses primarily change along one axis. 1D elements are commonly used in bridge analyses, frame structures, trusses, and other constructions where simplifying the model to one dimension significantly reduces the number of degrees of freedom, thus shortening computation time.
• 2D Elements: Two-dimensional elements, such as shells, membranes, and flat elements, are used in the analysis of surface structures. Shells are used in modeling thin-walled structures like walls, plates, pressure vessels, and vehicle bodies. Membranes, on the other hand, are used in stress analysis in flexible structures where tensile forces are the main loads. 2D elements allow for more detailed representation of stress and strain states than 1D elements but with fewer degrees of freedom than 3D elements.
• 3D Elements: Volumetric elements, such as solids, are used for full spatial analysis of three-dimensional structures. They are used where the structure has comparable dimensions in all three axes and where detailed representation of stress and strain distribution throughout the material volume is required. Examples of applications include analysis of mechanical components like gears, machine housings, aerospace elements, and other complex spatial structures.

Convergence Criteria and Evaluating Mesh Size in FEM

Choosing the right mesh size in FEM models is crucial for obtaining accurate and reliable results, but how can one assess whether the mesh size is appropriate? One of the most important tools for this purpose is convergence criteria, which allow engineers to evaluate whether the calculation results are approaching real values and whether further mesh refinement will bring significant benefits.

Convergence in the context of FEM refers to the convergence of results as the number of mesh elements increases (or the element size decreases). In practice, convergence criteria can be assessed in several ways:

1. Displacement Convergence: One of the basic indicators is the stabilization of displacement values at key points in the model. If displacements do not change significantly after further mesh refinement, it indicates that the mesh is already sufficiently accurate.
2. Stress Convergence: Similarly, the stabilization of stress distribution in critical areas of the model is analyzed. Convergence can be assessed by comparing stresses obtained for increasingly dense meshes – when differences become minimal, it indicates that the mesh is appropriate.
3. Internal Energy: The deformation energy of the system is another indicator that can be used to assess convergence. If the sum of internal energy stops changing significantly with further mesh refinement, it can be assumed that the mesh has achieved the appropriate accuracy.
4. Iterative Criterion: In iterative methods, such as nonlinear analysis, convergence can be assessed based on the number of iterations needed to reach a solution. If the number of iterations stabilizes with subsequent mesh refinements, it indicates that the model is approaching the true solution.
5. Comparison with Analytical or Experimental Solution: In some cases, FEM results can be compared with results obtained from analytical equations or experimental data. If the differences are small and further mesh refinement does not lead to significant improvement in agreement, the mesh is sufficient.

Singularities in FEM Calculations: The Problem of Infinite Stresses with Mesh Refinement

Mesh refinement in the Finite Element Method (FEM) aims to improve result accuracy, but it can lead to certain problems, especially in areas of singularities, such as sharp edges, corners, or stress concentration points. In these areas, as the mesh becomes finer, stresses can theoretically rise to infinite values, a known issue in FEM calculations.

Singularities in FEM occur when the geometric model contains points or lines where theoretically stresses can reach infinite values. Examples include sharp corners, gaps, or contact points between different materials. When the mesh is refined in these areas, local stresses can significantly increase, often exceeding values realistically observable in real materials. This phenomenon arises from the fact that FEM, as a numerical method, attempts to approximate the solution to the problem, but in areas with extreme stress gradients, it can lead to distorted results.

How to Address the Problem of Infinite Stresses?

1. Geometry Regularization: One approach to avoid infinite stresses is to modify the model’s geometry by rounding sharp edges or corners. While this may slightly alter the model, it allows for more realistic results without unjustified stress spikes.
2. Adaptive Mesh: Another approach is using adaptive meshes that automatically adjust element size in areas with high stress gradients. Instead of infinitely refining the mesh, larger elements are used in less critical areas and finer ones in high-stress areas, allowing for better representation of real conditions.
3. Stop Criteria: During convergence analysis, criteria can be set to stop further mesh refinement when stresses begin to exhibit nonlinear growth without proportional accuracy improvement in other parts of the model. This approach balances analysis accuracy and realism.
4. Scaling Analysis: In engineering practice, stress scaling is often used to assess their impact on the structure. Instead of assuming absolute values, stress proportions and distribution are analyzed to identify potential issues without precisely determining extreme values.

The above methods help minimize the problem of infinite stresses while ensuring that the FEM model remains realistic and useful for further analyses. It is particularly important for engineers to be aware of potential pitfalls associated with mesh refinement and apply appropriate techniques to avoid them.

FEM Calculations: Techniques for Improving Convergence

1. Adaptive Meshing: One technique for enhancing convergence effectiveness is adaptive meshing, which automatically adjusts mesh density based on the model’s local needs. This way, the mesh is refined only where truly necessary, saving computation time while maintaining high accuracy.
2. P-Refinement: This involves increasing the polynomial degree used for approximation in mesh elements instead of refining the mesh itself (h-refinement). It is particularly useful in analyses where there are areas with significantly varying stress gradients.
3. Method Hybridization: In some cases, a combination of different methods, such as the finite element method with the finite difference method or spectral element method, is used. This approach allows for better convergence in complex models.
4. Error Analysis: Conducting error analysis at various stages of mesh refinement allows for assessing whether further refinement is justified. If the computational error becomes insignificant, further mesh refinement may be unnecessary.

Applying appropriate criteria and techniques for improving convergence is crucial for obtaining reliable results in FEM calculations. Understanding when the model has achieved adequate convergence allows for optimizing the computational process, reducing analysis time, and ensuring that results align with reality.

FAQ: FEM Calculations