Posted by: abdurrosyid | February 21, 2012

FEM dan Metode Numerik PDE Lainnya

Finite Element Analysis (FEA) atau Finite Element Method (FEM) merupakan salah satu metode numerik untuk menyelesaikan persamaan diferensial parsial (PDE). Persamaan diferensial sendiri merupakan model matematika yang merepresentasikan nilai suatu variabel yang berubah terhadap variabel yang lainnya. Jika variabel tersebut berubah terhadap lebih dari satu variabel lainnya, persamaan diferensial biasa tidak memadahi, dan karenanya harus menggunakan persamaan diferensial parsial. Misalnya, perubahan besarnya stress terhadap posisi dalam sumbu x dan y, atau dalam sumbu x, y, dan z. Atau perubahan besarnya stress terhadap posisi dalam sumbu x, y, dan z dan juga terhadap waktu t.

Selain FEM, terdapat pula beberapa metode numerik lainnya untuk memecahkan persamaan diferensial parsial, seperti Finite Difference Method (FDM) dan Finite Volume Method (FVM). Ada juga Boundary Element Method (BEM) yang mirip dengan FEM, hanya saja diskritisasi hanya dilakukan pada boundary (surface) saja.

Situs menulis tentang perbedaan antara FEM, FDM, dan FVM sebagai berikut:

A finite difference method (FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula (that can generally be derived from a Taylor series expansion). The computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements. The discretization results in a system of equation of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.

A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy). The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM. The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh in to structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centers rather than at nodal points. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are obtained using interpolation.

A finite element method (FEM) discretization is based upon a piecewise representation of the solution in terms of specified basis functions. The computational domain is divided up into smaller domains (finite elements) and the solution in each element is constructed from the basis functions. The actual equations that are solved are typically obtained by restating the conservation equation in weak form: the field variables are written in terms of the basis functions, the equation is multiplied by appropriate test functions, and then integrated over an element. Since the FEM solution is in terms of specific basis functions, a great deal more is known about the solution than for either FDM or FVM. This can be a double-edged sword, as the choice of basis functions is very important and boundary conditions may be more difficult to formulate. Again, a system of equations is obtained (usually for nodal values) that must be solved to obtain a solution.

Comparison of the three methods is difficult, primarily due to the many variations of all three methods. FVM and FDM provide discrete solutions, while FEM provides a continuous (up to a point) solution. FVM and FDM are generally considered easier to program than FEM, but opinions vary on this point. FVM are generally expected to provide better conservation properties, but opinions vary on this point also. If you are trying to decide which method to use, then the best path is probably found by consulting the literature in the specific problem area.

Diantara FDM, FVM, dan FEM, barangkali yang bersaing lebih ketat adalah antara FVM dan FEM. Dalam pemakaian di lapangan, FVM sepertinya lebih banyak dipakai untuk fluida, sementara FEM lebih banyak dipakai untuk solid. Adapun FDM bisa dibilang memiliki lebih banyak kekurangan dibanding yang lainnya. Beberapa artikel menulis bahwa FDM kurang bagus untuk obyek yang geometrinya rumit. Juga, kurang cocok untuk diskritisasi yang tidak uniform.

Dalam FEM, model didiskritisasi menjadi sekian banyak finite element. Sedangkan dalam FVM, model didiskritisasi dalam sekian banyak control volume dimana computational node terletak di tengah-tengah control volume.

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