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. 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. 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.
This book o?ers a fundamental and practical introduction to the method, its variants, and their applications. In presenting the material, I have attempted to. This book serves as a text for one- or two-semester courses for upper-level undergraduates and beginning graduate students and as a professional reference for.
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. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.
Source Commun. Zentralblatt MATH identifier Efendiev, Y.
More by Y. More by T. Zhuming Bi, Ph. He holds two Ph.
Before moving to the US in , Dr. His research interests include finite element analysis, modelling and simulation, machine designs, robotics and automation, and enterprise systems. Bi has authored or co-authored 7 book chapters, 83 academic articles in international journals, and 45 papers in conference proceedings in these research areas.
He has over 20 years of teaching and research experience in Finite Element Analysis. We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website.
Thanks in advance for your time. Skip to content. Search for books, journals or webpages All Pages Books Journals. View on ScienceDirect. Authors: Zhuming Bi. Paperback ISBN: Imprint: Academic Press. Published Date: 4th January Page Count: