Question 1. What Is The Finite Element Method (fem)?
The FEM is a singular numerical approach used to clear up regular and partial differential equations. The technique is based on the combination of the phrases inside the equation to be solved, in lieu of point discretization schemes like the finite distinction technique. The FEM makes use of the method of weighted residuals and integration by elements (Green-Gauss Theorem) to lessen second order derivatives to first order phrases. The FEM has been used to resolve a huge variety of problems, and permits bodily domain names to be modeled at once using unstructured meshes commonly primarily based upon triangles or quadrilaterals in 2-D and tetrahedrons or hexahedrals in three-D. The solution area is discretized into character factors – those elements are operated upon personally after which solved globally using matrix solution strategies.
Question 2. What Is The History Of The Fem?
Early paintings on numerical solution of boundary-valued troubles can be traced to using finite distinction schemes; South well used such techniques in his e-book published inside the mid 1940’s. The beginnings of the finite detail approach genuinely stem from those early numerical techniques and the disappointment related to attempting to use finite distinction techniques on more hard, geometrically irregular issues. Beginning inside the mid Fifties,efforts to resolve continuum troubles in elasticity using small, discrete "elements" to explain the overall conduct of simple elastic bars began to appear, and such techniques had been to start with carried out to the plane industry. Actual coining of the time period "finite element"appeared in a paper with the aid of Clough in 1960. The early use of finite factors lay in the software to structural-related problems. However, others quickly identified the versatility of the approach and its underlying wealthy mathematical foundation for utility in non-structural regions. Since those early works, fast boom in usage of the technique has persisted since the mid Seventies. Numerous articles and texts had been posted, and new programs appear automatically within the literature.
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Question 3. What Is The Method Of Weighted Residuals, I.E., Galerkin’s Method?
The underlying mathematical foundation of the finite element method first lies with the classical Rayleigh-Ritz and variational calculus strategies. These theories furnished the reasons why the finite element technique worked nicely for the magnificence of problems in which variational statements can be obtained (e.G., linear diffusion type troubles). However,as hobby accelerated in applying the finite detail method to extra sorts of problems, the usage of classical idea to describe such issues became limited and could not be carried out, e.G., fluid-related troubles. Extension of the mathematical basis to non-linear and non-structural troubles became completed via the method of weighted residuals (MWR), in the beginning conceived by way of Galerkin in the early twentieth century. The MWR turned into located to provide the ideal theoretical basis for a much wider basis of troubles rather than the Rayleigh-Ritz approach. Basically, the technique calls for the governing differential equation to be improved through a hard and fast of predetermined weights and the resulting product incorporated over area; this necessary is needed to vanish. Technically, Galerkin's technique is a subset of the general MWR technique, seeing that diverse types of weights can be utilized; within the case of Galerkin's method, the weights are selected to be the same as the features used to outline the unknown variables. Most practitioners of the finite element approach now rent Galerkin's method to set up the approximations to the governing equations.
Question 4. Why Should One Use Finite Elements?
The versatility of the FEM, along side its rich mathematical formula and robustness makes it a super numerical method for a wide range of problems. The capability to model complicated geometries the use of unstructured meshes and using factors that may be individually tagged makes the approach unique. The ease of imposing boundary conditions as well as being able to use a wide circle of relatives of detail kinds is a precise gain of the scheme over different strategies. In addition, the FEM may be proven to stem from well-posed functional minimization ideas.
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Question five. Can The Fem Handle A Wide Range Of Problems, I.E., Solve General Pdes?
While the FEM was initially advanced to resolve diffusion kind troubles, i.E., stress-pressure equations or heat conduction, advances during the last numerous many years have enabled the FEM to solve advection-dominated troubles, which includes incompressible as well as compressible fluid drift. Modifications to the fundamental process (utilising types of upwinding for advection, i.E., Petrov-Galerkin and adaptive meshes) allow widespread advection-diffusion transport equations to be correctly solved for a huge variety of problems.
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Question 6. What Is The Advantage Of The Fem Over Finite Difference (fdm) And Finite Volume (fvm) Methods?
The fundamental benefits of the FEM over FDM and FVM are its integrated competencies to deal with unstructured meshes, a wealthy family of element selections, and herbal managing of boundary situations (particularly flux members of the family). The FDM is usually restrained to easy geometries in which an orthogonal grid can be built; for abnormal geometries, a international transformation of the governing equations (e.G., boundary geared up coordinates) ought to be made to create an orthogonal computational area. Likewise, implementation of boundary situations in FDM can be cumbersome. The FVM is an fundamental method (typically with limits -0.Five to 0.5) similar to the FEM, with volumes being used in preference to factors. The divergence theorem is used to set up the very last equation set. Solutions are acquired at quantity faces, vertices, or quantity centers – some methods rent staggered grids. While FVM can deal with irregular domains the use of unstructured grids (stemming from the FEM), the required averaging over the quantity limits the technique to 2d order spatial accuracy.
Question 7. Is There Any Connection Between The Fem And The Boundary Element Method (bem)?
In the BEM, one reduces the order of the hassle via one, i.E., a two-dimensional domain is decreased to a line vital – a three-dimensional domain will become a two-dimensional surface. The BEM best calls for the discretization of the bounds of the hassle area – no inner meshing is required, as inside the FDM, FVM, and FEM schemes. The BEM calls for two programs of the Green-Gauss Theorem (as opposed to one in the FEM and employing Galerkin’s Method). The approach is ideal for coping with abnormal shapes and setting boundaries that can volume to (close to) infinity. One can location indoors nodes within the BEM to reap internal values easily. The BEM works pretty successfully for linear differential equations – mainly elliptic equations. However, if one wants to clear up nonlinear advection-diffusion delivery equations, the method will become very cumbersome and computationally traumatic – BEM matrices are dense, and do no longer without difficulty permit efficient, sparse matrix solvers to be used as inside the FEM.
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Question 8. What Is Adaptivity, I.E., H-, P-, R-, And Hp-adaptation?
Adaptivity is an lively research place regarding either remeshing or increased interpolation order at some stage in the answer procedure. The method is in particular effective in fluid float, heat switch, and structural evaluation. The use of mesh refinement has been mainly effective in aerodynamic simulations for appropriately shooting shock places in compressible float. Generally, there are two kinds of edition: h-edition (mesh refinement), in which the element size varies at the same time as the orders of the form capabilities arekept regular; p-edition, wherein the detail length is constant while the orders of the shape capabilities are expanded (linear, quadratic, cubic, and so forth.). Adaptive remeshing (known as r-edition) employs a spring analogy to redistribute the nodes in an present mesh -no new nodes are delivered; the accuracy of the solution is constrained via the initial wide variety of nodes and factors. In mesh refinement (h-variation), character factors are subdivided with out altering their authentic position. The use of hp-model consists of each h- and p-version strategies and produces exponential convergence prices. Both mesh refinement and adaptive remeshing are now robotically used in lots of commercial codes. A spectral detail is a special class of FEM that uses a series of orthogonal foundation functions whereby the unknown phrases are solved at selected spectral nodes; the method is stable and particularly accurate, but can come to be time ingesting.
Question nine. How Difficult Is It To Write A Fem Program?
Writing a FEM code is not extraordinarily hard, specially if one develops the code using a general set of subroutines, e.G., enter data, integration, assembly, boundary situations,output, etc. About 90% of a FEM software is commonplace, which is fairly commonplace among most FEM codes – they generally tend to apply similar matrix solvers, quadrature guidelines, and matrix assembly strategies; I/O is normally the predominant difference among business FEM codes – some are clean, and a few are not so clean to research and use. A supply listing of the FORTRAN codes may be observed in the FORTRAN report folder; float charts may be obtained from the authors. Likewise, MATLAB and MathCad files also are to be had. One of the exceptional commercial programs now in the marketplace is COMSOL, which also allows users to put in writing their own solver programs and PDEs.
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Question 10. Are There Any Recommended Commercial Fem Packages That Are Versatile In Handling A Wide Range Of Problems?
Any of the widely recognized and extensively flexible FEM codes now in the marketplace are appropriate – it simply relies upon on how at ease the person is with the I/O part of this system. COMSOL,as referred to before, is pretty clean and very flexible – handling a wide variety of problem training consisting of fluid glide (with turbulence), heat switch, structural analysis,electrodynamics, and widespread PDEs along with species delivery, chemical reactions, and groundwater/porous media flows.
Question 11. Any Suggested Web Sites For Fem?
There are several endorsed web web sites:
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Question 12. How Long Does It Take For Me To Be Able To Use A Fem Program?
Some applications can help you remedy issues fairly quick. It is continually exceedingly encouraged that training session the example issues commonly provided by using maximum business software program. COMSOL, ANSYS, ALGOR, and NASTRAN all run on PCs.
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Question thirteen. Why Would I Want To Use A Fem Program?
The versatility, ease of statistics enter, and answer accuracy make the FEM one of the first-rate numerical methods for solving engineering problems. FEM programs are the spine of structural analyses, and are getting more broadly accepted for troubles in which geometries are complex.
Question 14. Is This A Method That Will Soon Become Obsolete?
The recent creation of BEM and meshless strategies would seem to signify the eventual obsolescence of the FEM. However, those more moderen methods are nevertheless years away from being evolved to the factor of extensive unfold applicability discovered in FEM. The FEM will be round for decades to return. Recent advances with the inclusion of spectral schemes and adaptivity make it particularly attractive now.
Question 15. How Expensive Is A Fem Code?
FEM codes variety from those that can be determined without spending a dime at the web to others costing many heaps of greenbacks. Those that run on PCs are generally cheaper, yet offer powerful gear for solving some of big scale issues.
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Question 16. What Kind Of Hardware Do I Need To Run A Fem Code?
A PC with a sufficiently fast processor, at least 256MB RAM, and at the least 20GB of difficult disk will allow many problems to be solved that when ought to best be run on mainframe computer systems. A counseled PC stage for main FEM calculations is one with 1 GB RAM, 60 GB hard disk, and walking with Pentium 4/3.2 GHz or better processors might offer greater that ok talents. The modern-day in PC hardware is enhancing constantly; in a few years, even these suggested requirements will seem out of date.