For analysis of engineering structures like geometrically nonlinear beams and plates in statics, I consider two main approaches: finite element method (FEM) and finite difference method (FDM). Since I am just starting, could you please explain me:
1. if I use FEM, is it possible to obtain a discretized system of the form Ax=b, where x is the displacement of the structure? As I understand, FEM reduces the problem to Kx=f, where K is the stiffness matrix and x is the coefficients of the approximation of the displacement with ansatz function. Am I right?
2. What are the advantages of FDM over FEM especially for nonlinear problems?
Traditionally, FEM more easily adapts to irregular structures. FEM also finds most frequent use in static systems (systems with no flow field) for both transient and steady-state analyses. FDM is most common for flowing systems that have convective terms as in computational fluid mechanics. Both approaches reduce to linear-algebra-type formulations that are commonly used by solvers in their solution. Frequently, efficient solvers, such as the Thomas algorithm, solve the system by moving some of the stiffness matrix terms back to the right side of the equation to form a tri-diagonal matrix. The nonlinearities of the system appear as terms involving X in the stiffness matrix, to use your terminology.
Dear Larry, thanks a lot for your answer. It is very useful.
FDM dealing usually with regular meshes and simple domains.
Meshes in the FEM are generally irregular and used for complex areas.
FEM is usually implicit method.
FDM can be explicit or implicit method.
Available explicit scheme on a triangular meshes, but it can not be called FEM.
Dear Professor Aptukov, that`s really what I need. Thank you very much.
I agree with previous comments. However, for PDEs on irregular domains you may also want to try using meshless methods using radial basis functions (RBFs). This approach can often be easier to program than FEM and gives comparable results. It also reduces to linear algebra type solutions and can be used for steady-state and dynamic problems. Fasshauer's book: Meshless Approximation Methods with Matlab, World Scientific Publishing Co., provides an excellent introduction along with code examples. Chapter 7 of my book: Numerical Analysis using R, Cambridge University Press,also provides an introduction with many references and code examples, see link for further information and some animated graphical solutions.
I hope this is useful.
http://www.pdecomp.net
Dear Prof. Griffits, thank you very much for your answer. It is very useful, but the link does not work.
Dear Prof. Griffits, your works are a reference point in numerical analysis.
Really, i am very interested with the answer of Valery Nagimovich Aptukov
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