By
JC Sun
December 28, 2022
Structural Analysis Finite Element Geotechnical Analysis midas FEA NX Meshing Finite Element Analysis stiffness solid element Detailed analysis blog Tetrahedron Pentahedron Hexagon
To ensure stable analysis performance and reliable result approximation, meshing is important. Meshing is the process to create finite elements and to connect those elements to formulate a set of functions. Finite elements are created by separating the known geometry with imaginary lines, and the elements are then connected with each other by specifying nodal connectivity at the element boundaries. Every element can be represented by a set of matrices (shown later), and connecting the elements essentially compiles the individual matrices into one structural matrix. Any non-time-dependent finite element analysis contains the following steps,
- Meshing
- Assigning boundary conditions
- Applying loads
- Numerical analysis
- Postprocessing
To ensure stable analysis performance and reliable result approximation, meshing is important. Meshing is the process to create finite elements and to connect those elements to formulate a set of functions. Finite elements are created by separating the known geometry with imaginary lines, and the elements are then connected with each other by specifying nodal connectivity at the element boundaries. Every element can be represented by a set of matrices (shown later), and connecting the elements essentially compiles the individual matrices into 1 structural matrix.
- Types of Finite Elements
Any non-time-dependent finite element analysis contains the following steps, Based on the shapes, there are the following types of finite elements:
- One-dimensional elements
- Two-dimensional elements
- Three-dimensional elements
In this article, we are going over each type of the finite element listed above, explaining its pros and cons, and when to use each type.
- One-Dimensional Elements
One-dimensional elements (line elements) include bar elements and beam elements, and the difference between the two depends on their load-bearing capabilities. Bar elements are suitable for modeling trusses because their load-bearing capabilities are limited in the axial directions. On the other hand, beam elements are suitable for modeling frames because they can resist bending, twisting, as well as axial forces. A beam that is continuous over two or more supports can usually be modeled using one beam element per span between supports without using several beam elements to model one individual span. Thus, when a one-dimensional element is used for static analysis, the discretization phase of modeling becomes trivial, and for stress analysis, the name “matrix methods of structural mechanics” may be used in preference to “FEA” [1].
To represent a bar in axial tension, shown in figure 1, the following considerations can be made to compile the stiffness matrix.
Figure 1. A prism bar with one end fixed and the other end subjected to a force P.
A prismatic bar of length L, cross-section area A, and elastic modulus E is subjected to a force P at one end and with the other end fixed. The tensile elongation of the bar can be represented as
For a two-node bar element, shown in figure 2,
Figure 2. A two-node bar element.
The equilibrium at each individual node gives the following:
Which can be written in the matrix format as:
further reduction gives:
Where [k] is the characteristic matrix (stiffness matrix), {d} is the displacement vector, and {r} is the loads associated with each individual node.
Figure 3. A beam with two nodes (a) with two degrees of freedom at each node, translational in y and rotation in z, (b) with vertical loads and rotational moments at each node.
Figure 3 shows a beam element with two nodes. Each node is subjected to two degrees of freedom (figure 3a) and two nodal forces (figure 3b). Using the Euler-Bernoulli beam theory, the following matrix equation can be formed:
The same element stiffness matrix can be obtained by calculating using interpolation and shape functions,
Where [B] is the strain-displacement matrix obtained from the shape functions [N]. We can also use shape function interpolation to obtain the stiffness matrix of 2D and 3D elements as well, as shown later in the article.
Line elements are widely used in general structural analysis to obtain an overview of structural behavior, as well as in the detailed analysis in conjunction with other types of elements as connections, stiffeners, etc. In bridge engineering, the general structural analysis using 1D elements gives engineers a comprehensive understanding of the bridge structural behavior, as well as provides member forces and moments for design code checking for various standards.
However, to obtain more insights about localized structural zones, or when analyzing complex bridge geometries, local refinement would be required. To satisfy those higher analysis requirements, two-dimensional elements and three-dimensional elements offer better result approximation. Figure 4 (left) shows the general analysis result using the 1D element, and figure 4 (right) shows the refined analysis for the high moment elements using 2D elements. The nodal moment loads in the localized detailed analysis are extracted from the general analysis results.
Figure 4. General analysis using 1D element and using its element force-moment results as initial loads for the 2D detailed analysis.
- Two-Dimensional Elements
To solve two-dimensional (2D) problems, 2D elements are needed. Similar to how stiffness matrix is constructed for 1D elements as shown in equation 6, 2D elements’ stiffness matrix can also be constructed from shape function and interpolation,
where again, [k] is the stiffness matrix, [B] is the strain-displacement matrix obtained from the shape functions, [E] is the constitutive matrix, t is the thickness of the element, and A is the area of the element.
Common 2D problems include plane stress, plane strain, shell, axisymmetric solid, geogrid 2D, and gauging shell elements. Plain strain and axisymmetric solid elements are 2D shape elements, but they are used to express 3D stress states [2]. The common 2D element shape used is the triangular element with 3 nodes, shown in figure 5a, this type of element is called a constant strain triangle (CST) because, in stress analysis, a linear displacement field produces a constant strain field [1]. The CST elements do not work very well, because the “locking” effect can make the mesh overly stiff [3]. Even though refining the mesh can help with the accuracy, it also does increase the analysis solving time. The inaccuracy due to “locking” can be improved with the 6 node triangular elements. As shown in figure 5b, they have middle nodes between the vertices. 6 node triangular elements are also known as linear strain triangles (LST) or quadratic triangles.
Figure 5. (a) Constant-strain triangle element, (b) Linear strain triangle element.
A simple but less used 2D element is the 4-node rectangular element (Q4) whose sides are parallel to the global coordinate systems. This system is easy to construct automatically but it is not well suited to approximate inclined boundaries [4]. The Q4 elements experience the same “locking” effects as the CST elements, however, the issue can be improved with quadratic rectangular elements (Q8, Q9) by adding mid edge nodes.
- Three-Dimensional Elements
As shown in figure 6(a), tetrahedron has 4 nodes and is the most basic 3D finite element. Figure 6(b) shows a pentahedral (pyramid) element, figure 6(c) shows an 8 node rectangular solid element (Q8), and figure 6(d) shows an 8 node hexahedral isoparametric element.
Figure 6. (a) 4 node tetrahedron element, (b) 5 node pentahedral (pyramid) element, (c) 8 node rectangular solid element, d) 8 node hexahedral isoparametric element.
Similar to rectangular Q4 elements and CST elements, Q8 also has the disadvantage of shear locking. Also similar to Q4 elements, Q8 rectangular solid elements are difficult to mesh irregular geometries, especially for geometries with varying mesh densities. To reduce shear locking, adding mid-edge nodes to the Q8 element would help, and making the elements isoparametric will help with meshing flexibilities. Isoparametric formulation permits quadrilateral and hexahedral elements to have non-rectangular shapes [1]. When combining the two efforts, adding mid-edge nodes to the linear isoparametric hexahedral elements would make them more versatile and produce better analysis results. 3D geometries are needed to perform 3D mesh and they come from CAD models, which take longer and more effort to produce. Furthermore, analysis containing 3D elements usually contains more nodes and elements thus would take longer to solve. 2D analysis, however, requires less time to mesh and less time to solve. The 2D analysis also provides a more sufficient amount of structural information than 1D analysis and produces results close to 3D analysis. However, 2D analysis can only replace 3D analysis when structures can be represented by plate elements. When a structure is more complex, as shown in figure 7, 3D elements need to be used.
Figure 7. A lug and pin model.
Engineers sometimes model structural elements using combined 1D/2D/3D mesh to take advantage of the benefit of each type of element while saving analysis time, as shown in the bridge model in figure 8.
Figure 8. A bridge model utilizing hybrid element types with merged nodes at the element boundaries
References:
[1] R. Cook, D. Malkus, M. Plesha, R. Witt, Concepts and Applications of Finite Element Analysis, Fourth Edition, 2001, John Wiley & Sons Inc., New York, NY.
[2] Midas Information Technology, Analysis Reference Midas FEA NX, Chapter 3, 2021.
[3] Th. Zimmermann, S. Commend, Stabilized Finite Element Applications in Geomechanics, Laboratory of Structural and Continuum Mechanics, Department of Civil Engineering Swiss Federal Institute of Technology, 2001, Lausanne-EPFL, Switzerland.
[4] S.S. Bhavikatti, Finite Element Analysis, 2005, New Age International Limited, New Delhi.
https://www.midasoft.com/bridge-library/civil/products/midasfeanx
jsun@midasoft.com
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FAQs
How do I know if my mesh is good enough? ›
The most basic and accurate way to evaluate mesh quality is to refine the mesh until a critical result such as the maximum stress in a specific location converges: meaning that it doesn't change significantly as the mesh is a refinement.
What is finite element mesh? ›The finite element mesh is used to subdivide the CAD model into smaller domains called elements, over which a set of equations are solved. These equations approximately represent the governing equation of interest via a set of polynomial functions defined over each element.
How does meshing work in FEA? ›Meshing for CFD and FEA facilitates accurate simulation of flow or other physical phenomena. Meshing discretizes a complex object into well-defined cells where the governing equation can be assigned so that the solver can easily simulate physical behavior.
How do I choose mesh size in FEA? ›- Perform chosen analysis for several different mesh sizes.
- Notice where high deformations or high stresses occur, perhaps it is worth to refine mesh in those regions.
- Collect data from analysis of each mesh: outcome, number of nodes in the model, computing time.
If your mesh is too coarse, the answers that you generate may not be as accurate as you need them to be. Conversely, if your mesh is too fine, the simulation will take much longer to run without providing any substantial improvement in accuracy.
What is finer 100 mesh or 200 mesh? ›...
Mesh and Micron Sizes.
| US Mesh* | |
|---|---|
| 35 | |
| 200 | |
| Microns | 74 |
| Inches | 0.0029 |
Finite element analysis (FEA) is the use of calculations, models and simulations to predict and understand how an object might behave under various physical conditions. Engineers use FEA to find vulnerabilities in their design prototypes.
What are types of meshing elements? ›The three types of meshing models are as follows: Tetrahedral - tetrahedral cell shape based core mesh. Polyhedral - polyhedral cell shape based core mesh. Trimmed - trimmed hexahedral cell shape based core mesh.
What is the most common FEA? ›- ANSYS. ANSYS is a popular FEA software that is extensively used in the engineering space. ...
- SimScale. SimScale is a cloud-based FEA software that serves as an alternative to ANSYS. ...
- Autodesk. ...
- ABAQUS. ...
- OpenFOAM.
Meshing is often used in software-based simulation for Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD). It can significantly impact the accuracy of the simulation and the resources required to perform the simulation.
What is the basics of mesh analysis? ›
What is Mesh Analysis? The method in which the current flowing through a planar circuit is calculated. A planar circuit is defined as the circuits that are drawn on the plane surface in which there are no wires crossing each other. Therefore, a mesh analysis can also be known as loop analysis or mesh-current method.
What are the steps in mesh analysis? ›- Identify the meshes.
- Assign a current variable to each mesh, using a consistent direction (clockwise or counterclockwise).
- Write Kirchhoff's Voltage Law around each mesh. ...
- Solve the resulting system of equations for all loop currents.
- Solve for any element currents and voltages you want using Ohm's Law.
What is Mesh Size? U.S. Mesh Size (or U.S. Sieve Size) is defined as the number of openings in one square inch of a screen. For example, a 36 mesh screen will have 36 openings while a 150 mesh screen will have 150 openings.
Which is finer 40 mesh or 100 mesh? ›Larger particles were trapped above in the 40 mesh screen and smaller particles passed through the 100 mesh screen. As a result the larger particles were eliminated from the distribution by the 40 mesh screen and smaller particles were eliminated by the 100 mesh screen.
What does 200 mesh size mean? ›"200 mesh" is commonly employed as a label attached to air-separated mineral powders used in ceramics. Generally, it refers to the coarsest particles present, if the material is passed through a 200 mesh sieve only a small amount of residue should be present. Frits are generally 200 mesh.
Why is a finer mesh better? ›Generally, in a Finite Element Analysis (FEA), a finer mesh produces more accurate results. The smaller elements in a finer mesh can more accurately capture stress gradients across the element.
Does mesh size matter? ›Typically, the smaller the mesh size, the more accurate the solution as the designs are better sampled across the physical domains. The trade-off is that the higher the accuracy, the larger the simulations become and thus solve times are extended.
What is the rule of thumb for mesh size? ›The mesh edge length can vary, to be finer in areas where needed. For all mesh types 3 elements across where a thickness change. For a 3D mesh type, the rule of thumb is a tetra edge length on surface 1-2x the thickness.
What mesh count do I need? ›Generally, mesh counts ranging from 25 mesh to 305 mesh are ideal when screen printing. Mesh count is the number of openings within an inch in any direction. This means a 305 mesh screen is much finer than a 25 mesh screen.
What is 160 mesh used for? ›160 Mesh - Our most popular mesh count. Works well with a wide variety of ink types and substrates. 200-230 Mesh - Most commonly used for high detail artwork. Also great for printing with water based inks on wood or paper.
What is ideal mesh size? ›
The basic mesh size of analysis models is 20 cm for the edges of the 4 nodes square shell elements. Smaller mesh size is used where large deformation is anticipated because of local buckling, i.e. at the welded joint of two sections with different thicknesses and at the base of the pier.
Can you learn FEA on your own? ›One of the most frequently asked questions by beginners to engineering simulation is how to learn finite element analysis, and how to use FEA software. This process is not easy, particularly if you want to learn by yourself, not in university. However, with a little motivation and direction, it is achievable.
What is difference between FEM and FEA? ›Engineers use FEM when they need to develop an adoptable design that's practical but not necessarily perfect for a particular application. FEA: The mathematical equations behind FEM are applied to create a simulation, or what's known as a finite element analysis (FEA).
How do you choose a meshing type? ›For simple geometries, use quadrilateral/hexahedral meshes. For moderately complex geometries, use unstructured quadrilateral/hexahedral meshes. For relatively complex geometries, use triangular/tetrahedral meshes with prism layers. For extremely complex geometries, use pure triangular/tetrahedral meshes.
How do you do good meshing? ›- A Simplified and Clean Watertight Geometry. ...
- Deciding and Maintaining a Good General Grid Size. ...
- Increasing Mesh Fineness at Critical Areas. ...
- Boundary-Layer Refinement and Y+ ...
- Mesh Convergence Study.
Meshing also fails due to overlapping surfaces or bodies present in the component. Hence, always check any assembly or part file for intersecting surfaces or merge features, respectively.
What are the negatives for using FEA? ›Disadvantages include:
The method is approximate — the analysis is not performed on a real structure, but on a model of it. All the results (such as stresses, strains, or displacements) are approximated and the user cannot precisely estimate the difference between the obtained results and the real ones.
FEA Errors Modeling Errors in FEA
Is there a material linearity or nonlinearity? Is there a geometric linearity or nonlinearity? Large deformation or strains? In relation to contact definition, is it with friction or without friction?
Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI , depending on the complexity of the domain and the type of mesh desired.
Why is quality of meshing important? ›The quality of the meshing is very important since it will determine the possibility of solving the case and the computational speed with which it would be solved. The complexity of this process will largely depend on the type of simulation and the characteristics of the model.
How many equations are in a mesh analysis? ›
Explanation: According to the formula, the number of mesh equations = B-(N-1). Total branches = 5 and nodes = 4. Hence, the number of mesh equations = 5-(4-1) = 5 – 3 = 2.
What is the difference between loop and mesh? ›A loop is any closed path through a circuit where no node quite once is encountered. A mesh is a closed path during a circuit with no other paths inside it.
Which law is used in mesh analysis? ›The mesh current method is a network analysis technique where mesh (or loop) current directions are assigned arbitrarily, and then Kirchhoff's voltage law (KVL) and Ohm's law are applied systematically to solve for the unknown currents and voltages.
Which mesh is finer 200 or 400? ›They measure the holes in the mesh in microns, so the higher the number, the larger the holes. 100 micron mesh holes are 0.003925” compared to 600 micron mesh which is 0.023550”. 400 micron mesh has 0.015700” holes, and 200 micron mesh has 0.007850” holes.
Is 60 mesh or 40 mesh finer? ›| Mesh Number | Inches | Millimetres |
|---|---|---|
| 40 | 0.0165 | 0.4 |
| 45 | 0.0138 | 0.354 |
| 50 | 0.0117 | 0.297 |
| 60 | 0.0098 | 0.25 |
Higher mesh numbers = smaller particle sizes. It is very important to remember that mesh size is not a precise measurement of the mesh opening size.
What does 20 mesh size mean? ›| US Mesh | Micron | Inches |
|---|---|---|
| 20 | 850 um | 0.0331 |
| 24 | 690 um | 0.027 |
| 30 | 560 um | 0.022 |
| 36 | 485 um | 0.019 |
You most often see 13 mesh and 18 mesh in the US market, but there are so many mesh sizes. The mesh size correlates to the number of stitches in a square inch - so 13 mesh, has 13 stitches in one inch. Because of that, the larger the mesh size, the smaller the stitches.
What does 40 mesh size mean? ›If a material is described as -40 mesh, then 90% or more of the material will pass through a 40-mesh sieve (particles smaller than 0.420 mm).
What size is 400 mesh? ›Mesh is the number of openings in one linear inch of any sieve or screen. A 10 mesh sieve will have 10 openings and a 400 mesh sieve will have 400 openings in one linear inch.
What is a 400 mesh? ›
400 mesh is a fine size U.S. Standard mesh size with a 0.0015" (38µm) nominal sieve opening with a typical wire diameter of 0.03mm.
What does 110 mesh mean? ›Mesh size is measured by how many threads of mesh cross per square inch. A 110 mesh, for example, has 110 threads crossing per square inch. The higher the mesh count, the finer the threads and holes are in the screen.
What is considered a good mesh? ›A good quality mesh has a Jacobian ratio between 1 and 10 for the majority of its elements (90% and above). Create a Mesh Quality Plot to plot the Jacobian ratio of all elements. For most models, elements at regions of high curvatures have higher Aspect and Jacobian ratios.
How do you test mesh coverage? ›- Open the Google Home app .
- Tap Wi-Fi. Wifi devices. Points. Test mesh.
Meshing is one of the most important steps in performing an accurate simulation using FEA. A mesh is made up of elements which contain nodes (coordinate locations in space that can vary by element type) that represent the shape of the geometry.
How can I make my FEA more accurate? ›The key to generating effective FEA meshes is to strike an appropriate balance between order and size for the particular problem that is being analyzed. When possible, use second-order elements and iteratively refine the mesh until the results converge.
What must be avoided in a mesh network? ›- fully connected mesh networks can be impractical to set up because of the high number of connections needed.
- many connections require a lot of maintenance.
- Run a speed test. ...
- Make sure you're connected to the 5Ghz band. ...
- Reposition your router. ...
- Update your Wi-Fi router's firmware. ...
- Switch to a less congested channel. ...
- Make sure there are no Wi-Fi freeloaders. ...
- Upgrade your router, or add extenders.
As an example, if you have a 2-story house that's 2,200 sq ft, we recommend 2 mesh points: one for the first (ground) floor, and one for the floor above (or below, in case of a basement).
Why is a smaller mesh better? ›Generally, in a Finite Element Analysis (FEA), a finer mesh produces more accurate results. The smaller elements in a finer mesh can more accurately capture stress gradients across the element.
Which is finer 200 mesh or 400 mesh? ›
400 micron mesh has 0.015700” holes, and 200 micron mesh has 0.007850” holes.