Do these formulations interest you? If yes, please read more in the forum topic about the equivalence between the weak and strong formulation of PDEs for FEA. In the case of one-dimensional elastostatics, the minimum of potential energy is resilient for conservative systems. Every infinitesimal disturbance of the stable position leads to an energetic unfavorable state and implies a restoring reaction. An easy example is a normal glass bottle that is standing on the ground, where it has minimum potential energy.
If it falls over, nothing is going to happen, except for a loud noise. If it is standing on the corner of a table and falls over to the ground, it is rather likely to break since it carries more energy towards the ground. For the variation principle, we make use of this fact. The lower the energy level, the less likely it is to get the wrong solution. Find more about the minimum potential energy in our related forum topic.
One of the most overlooked issues in computational mechanics that affect accuracy is mesh convergence. This is related to how small the elements need to be to ensure that the results of an analysis are not affected by changing the size of the mesh. The figure above shows the convergence of a quantity with an increase in the degrees of freedom. As depicted in the figure, it is important to first identify the quantity of interest. At least three points need to be considered and as the mesh density increases, the quantity of interest starts to converge to a particular value.
If two subsequent mesh refinements do not change the result substantially, then one can assume the result to have converged. Going into the question of mesh refinement, it is not always necessary that the mesh in the entire model is refined. Hence, from a physical point of view, the model can be refined only in particular regions of interest and further have a transition zone from coarse to fine mesh.
There are two types of refinements h- and p-refinement as shown in the figure above. Here it is important to distinguish between geometric effect and mesh convergence, especially when meshing a curved surface using straight or linear elements will require more elements or mesh refinement to capture the boundary exactly.
Mesh refinement leads to a significant reduction in errors:. Refinement like this can allow an increase in the convergence of solutions without increasing the size of the overall problem being solved. So now that the importance of convergence has been discussed, how can convergence be measured? What is a quantitative measure for convergence? The first way would be to compare with analytical solutions or experimental results.
As shown in the equations above, several errors can be defined for displacements, strains, and stresses.
These errors could be used for comparison and they would need to reduce with mesh refinement. Learn more about how these errors are calculated with the respective norms for these quantities here. The Finite Element Analysis started with significant promise in modeling several mechanical applications related to aerospace and civil engineering. The applications of Finite Element Method are just starting to reach their potential.
One of the most exciting prospects is its application to coupled problems like fluid-structure interaction; thermo-mechanical, thermo-chemical, thermo-chemo-mechanical problems piezoelectric, ferroelectric, electromagnetics and other relevant areas:.
Apply FEM yourself! Do you also feel like applying finite element methods to real life scenarios, like a running wheel, or a drone, or a phone dropped accidentally. All of this is possible by simply signing up as our Community or Professional SimScale user.
With static analysis, you can analyze linear static and nonlinear quasi-static structures. In a linear case with an applied static load, only a single step is needed to determine the structural response. Geometric, contact, and material nonlinearity can be taken into account. An example is a bearing pad of a bridge. Dynamic analysis helps you analyze the dynamic response of a structure that experienced dynamic loads over a specific time frame.
To model the structural problems in a realistic way, you can also analyze the impacts of loads as well as displacements. An example is the impact of a human skull , with or without a helmet. Eigenfrequencies and eigenmodes of a structure due to vibration can be simulated using modal analysis.
The peak response of a structure or system under a given load can be simulated with harmonic analysis. An example is the start of an engine. As discussed earlier in the section on PDEs, traditional FEM technology has demonstrated shortcomings in modeling problems related to fluid mechanics, wave propagation, etc. Several improvements have been made over the last two decades to improve the solution process and extend the applicability of finite element analysis to a wide genre of problems.
Some of the important ones still being used include:. The Bubnov-Galerkin method requires continuity of displacements across elements. To run an FEA simulation, a mesh is first generated, containing millions of small elements that make up the overall shape. This is a way of transcribing a 3D object into a series of mathematical points that can then be analyzed. The density of this mesh can be altered based upon how complex or simple a simulation is needed.
Calculations are run for every single element or point of the mesh and then combined to make up the overall final result for the structure. Since the calculations are done on a mesh, rather than the entirety of a physical object, it means that some interpolation needs to occur between the points.
These approximations are usually within the bounds of what's needed. The points of the mesh where the data is known mathematically are referred to as nodal points and tend to be grouped around boundaries or other areas of change in an object's design. For example, if you know the temperature at one point in an object, how would you determine the exact temperature at other points of the object, dependent upon time? Utilizing FEA, an approximation can be made for these points using different modes of accuracy.
There's a square approximation, a polynomial approximation, and a discrete approximation. Each of these techniques increases in accuracy and complexity.
If you're really interested in the intense mathematical side of FEA, take a look at this post from SimScale that goes into the nitty-gritty. The other type of FEA that we mentioned earlier is Computational Fluid Dynamics, which warrants a look into how it's used. In the early s, scientists and engineers were already using these equations to solve fluid problems, but due to the lack of computing power, the equations were simplified and reduced to 2 dimensions.
While rudimentary, these first practical applications of fluid dynamic analysis gave way to what would soon be an essential simulation asset. For most of the early years, solving CFD problems entailed simplifying equations to the point that they could be done by hand. By no means was the average engineer using these calculations; rather, up until the late s, CFD remained a largely theoretical and exploratory practice. Tealium We use Tealium to collect data about your behavior on our sites.
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