Based on the type of PDE, the necessary inputs can be evaluated. FEA falls into the second category of variational methods. Variational approaches are primarily based on the philosophy of energy minimization. The wave equation for instance is a hyperbolic PDE. However, over the years, modifications have been developed to extend the applicability of FEA technology.

It is important to consider the consequence of using a numerical framework that is unsuitable for the type of PDE that is chosen. This could mean that small changes in the domain parameters lead to large oscillations in the solutions or the solutions exist only on a certain part of the domain or time. These are not reliable. Well-posed solutions are defined with a unique one, that exists continuously for the defined data.

Hence, considering reliability, it is extremely important to obtain them. The mathematical models of heat conduction and elastostatics covered in this series consist of partial differential equations with initial conditions as well as boundary conditions. This is also referred to as the so-called Strong Form of the problem. That means that the second derivative of the displacement has to exist and has to be continuous! To develop the finite element formulation, the partial differential equations must be restated in an integral form called the weak form.

The weak form and the strong form are equivalent! In stress analysis, the weak form is called the principle of virtual work.

## Basic Finite Element Analysis (FEA)

The given equation is the so-called weak form in this case the weak formulation for elastostatics. The name states that solutions to the weak form do not need to be as smooth as solutions of the strong form, which implies weaker continuity requirements.

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You have to keep in mind that the solution satisfying the weak form is also the solution of the strong counterpart of the equation. This is an essential property of the trial solutions and this is why we call those boundary conditions essential boundary conditions. If you want to read about the equivalence between the weak and strong formulation, please read more in the forum topic about the equivalence between the weak and strong formulation of PDEs for FEA. The Finite Element Analysis can also be executed with a variation principle.

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 that the bottle breaks 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 a 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.

Figure 4: Convergence of Quality with increasing Degrees of Freedom. The figure above shows the convergence of quantity with increase in 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:. Figure 6: Practical application of Mesh Refinement. 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 displacement, strains and stresses. These errors could be used for comparison and they would need to reduce with mesh refinement. However, in a FEA mesh, the quantities are calculated at various points nodal and Gauss.

So in this case, what needs to be determined is where and at how many points should the error be calculated. Learn more about how to measure convergence.

### What is Strain?

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:.

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.

## FEA For Beginners – Finite Element Analysis | SimScale Blog

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 impacts of loads as well as displacements.

An example is the impact of a human skull , with or without a helmet. Discretization 5. Discretized equations 5. Matrix solution for linear elastic problems 5. Numerical integration 5.

## Finite element method

Linear elastic patch tests 5. Application 1: pure bending of a linear elastic beam 5. Application 2: square domain with circular hole 5.

Mixed approach to nonlinear problems 5. Step-by-step solution of the discretized nonlinear equations 5. Example of an elastoplastic material 5. Application: pure bending of an elastoplastic beam 5. Conclusion Chapter 6. Flow Models 6. Natural element method in fluid mechanics: updated Lagrangian approach 6. Free and moving surfaces 6. Short-fiber suspensions flow 6. Breaking dam problem 6. Multi-scale approaches Chapter 7. Conclusion Bibliography Index Du kanske gillar.

Lifespan David Sinclair Inbunden. Inbunden Engelska, Spara som favorit. Skickas inom vardagar. Laddas ned direkt. Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics.

Before the emergence of computational science also called scientific computing as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science. This book presents a recent state of the art on the foundations and applications of the meshless natural element method in computational mechanics, including structural mechanics and material forming processes involving solids and Newtonian and non-Newtonian fluids. Passar bra ihop.