Finite element method

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The finite element method ( FEM ) or finite element analysis ( FEA ), is a numerical method for solving engineering problems and mathematical physics. Common problem areas include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of this problem generally requires a solution to the problem of boundary values ​​for partial differential equations. The finite element method formulation of the problem produces a system of algebraic equations. This method generates an approximate value of the unknown on a number of discrete points over the domain. To solve the problem, it divides the big problem into smaller and simpler parts called up to elements. A simple equation modeling the finite element is then assembled into a larger system of equations that model the entire problem. FEM then uses the variational method of calculus variation to estimate the solution by minimizing the associated error function.

Video Finite element method

Basic concepts

The division of entire domains into simpler parts has several advantages:

• An accurate representation of complex geometries
• Inclusion of different material properties
• Easy representation of total solutions
• Capture local effects.

A typical work of this method involves (1) dividing the domain of the problem into a subdomain set, with each subdomain represented by a set of element equations for the original problem, followed by (2) systematically combining all sets of equation elements into the global system of equations for final calculation. The system of global equations has known the solution techniques, and can be calculated from the initial value of the original problem to obtain numerical answers.

In the first step above, the element equation is a simple equation that locally approximates the original complex equation to be studied, where the original equation is often a partial differential equation (PDE). To explain the approach in this process, FEM is generally introduced as a special case of the Galerkin method. The process, in the language of mathematics, is to build the integral of the inner product of residual and heavy functions and set the integral to zero. In simple terms, this is a procedure that minimizes the error approach by functioning the experimental function into the PDE. Residuals are errors caused by experimental functions, and the weight function is a polynomial approximation function that projects residuals. The process removes all the spatial derivatives of the PDE, thus approaching the local PDE with

• a set of algebraic equations for the steady state problem,
• a set of ordinary differential equations for a temporary problem.

This equation set is the equation of the element. They are linear if the underlying PDE is linear, and vice versa. The set of algebraic equations that appear in the steady state problem is solved by using numerical linear algebraic methods, whereas the usual sets of differential equations appearing in transient problems are solved by numerical integration using standard techniques such as the Euler method or the Runge-Kutta method.

In step (2) above, the global system of equations is generated from the equation of the element through the coordinate transformation of the local subdomain node to the global domain node. This spatial transformation includes an appropriate orientation adjustment as applied in conjunction with a reference coordinate system. This process is often done by FEM software using coordinate data generated from subdomains.

FEM is best understood from its practical application, known as finite element analysis (FEA) . FEA as applied in engineering is a computational tool for performing technical analysis. This includes the use of mesh generation techniques to divide complex problems into small elements, as well as the use of software programs encoded with FEM algorithms. In applying FEA, complex problems are usually physical systems with underlying physics such as Euler-Bernoulli beam equations, heat equations, or Navier-Stokes equations expressed in PDE or integral equations, whereas small elements are divided from complex problems representing different areas within physical systems.

FEA is a good choice for analyzing issues over complex domains (such as cars and oil pipelines), when domains change (as during solid state reactions with moving limits), when desired precision varies across domains, or when a solution does not smoothness. FEA simulations provide valuable resources because they remove some examples of hard prototyping manufacture and testing for various high fidelity situations. For example, in a simulated frontal collision it is possible to improve prediction accuracy in "important" areas such as the front of the car and reduce it behind (thus reducing the cost of simulation). Another example is in numerical weather forecasts, where it is more important to have accurate predictions about the development of nonlinear phenomena (such as tropical cyclones in the atmosphere, or ocean vortices) rather than relatively quiet areas.

Maps Finite element method

History

Although it is difficult to cite the date of the discovery of the finite element method, this method derives from the need to resolve complex elasticity and the problem of structural analysis in civil engineering and aeronautics. Its development can be traced back to the work of A. Hrennikoff and R. Courant in the early 1940s. Another pioneer is Ioannis Argyris. In the USSR, the introduction of the practical application of this method is usually associated with the name Leonard Oganesyan. In China, in the 1950s and early 1960s, based on dam construction calculations, K. Feng proposed a systematic numerical method for solving partial differential equations. This method is called a different method up to the principle of variation, which is another independent discovery of the finite element method. Although the approach used by these pioneers is different, they share one important characteristic: discrete mesh from a continuous domain into a set of separate sub-domains, usually called elements.

Hrennikoff's work discretizes the domain using lattice analogies, while the Courant approach divides the domain into subregional triangles to solve second-order iptic partial differential equations (PDE) arising from the cylinder torque problem. Courant's contribution is evolutionary, drawing on large bodies of initial results for PDE developed by Rayleigh, Ritz, and Galerkin.

The finite element method got a real boost in the 1960s and 1970s by the development of JH Argyris with colleagues at the University of Stuttgart, RW Clough with colleagues at UC Berkeley, OC Zienkiewicz with Ernest Hinton colleagues Bruce Irons and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with colleagues at Cornell University. Further encouragement is given in these years by the open source finite element software programs available. NASA sponsors the original version of NASTRAN, and UC Berkeley makes the program elements up to SAP IV widely available. In Norway, the ship classification society Det Norske Veritas (now DNV GL) developed Sesam in 1969 for use in ship analysis. The strict mathematical basis for the finite element method was given in 1973 with the publication by Strang and Fix. This method has been generalized for numerical modeling of physical systems in various disciplines, such as electromagnetism, heat transfer, and fluid dynamics.

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Technical discussion

The finite element method structure

The finite element method is a numerical method to approximate the solution of a mathematical problem that is usually formulated so as to precisely express the idea of ​​some aspect of physical reality.

The finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures.

Examples of variational formulations are Galerkin method, Galerkin method disconnected, mixed methods, etc.

The discretization strategy is understood to mean a clear set of procedures that include (a) the creation of elemental traps up, (b) the definition of the base function on the reference element (also called the function of the form) and (c) mapping the reference element to the mesh element. Examples of discrete strategies are h-versions, p-versions, hp-versions, x-FEMs, isogeometry analysis, etc. Each discrete strategy has certain advantages and disadvantages. A sensible criterion in choosing a discrete strategy is to realize an almost optimal performance for the most extensive set of mathematical models in a particular model class.

There are various numerical solution algorithms that can be classified into two broad categories; direct and iterative solver. The algorithm is designed to exploit the sparsity of the matrix which depends on the choice of variational formulation and discretization strategies.

The postprocessing procedure is designed for the extraction of interesting data from finite element solutions. To meet the requirements of the verification solution, postprocessors must provide an estimate of error a posteriori in the case of the amount of interest. When the approach error is greater than what is deemed acceptable then discretization must be changed either by an automatic adaptive process or by analyst action. There are some very efficient postprocessors that provide for the realization of superconvergence.

Illustrative Problems P1 and P2

We will show the finite element method using two examples of problems from which common methods can be extrapolated. It is assumed that readers are familiar with linear calculus and algebra.

P1 adalah masalah satu-dimensi

${\ displaystyle {\ mbox {P1}}: {\ begin {cases} u '' (x) = f (x) {\ mbox {in}} (0, 1), \\ u (0) = u (1) = 0, \ end {cases}}}  ÂÂ$

di mana ${\ displaystyle f}  ÂÂ$ diberikan, ${\ displaystyle u}  ÂÂ$ adalah fungsi yang tidak diketahui dari ${\ displaystyle x}  ÂÂ$ , dan ${\ displaystyle u ''}  ÂÂ$ adalah turunan kedua dari ${\ displaystyle u}  ÂÂ$ sehubungan dengan ${\ displaystyle x}  ÂÂ$ .

P2 adalah masalah dua dimensi (masalah Dirichlet)

${\ displaystyle {\ mbox {P2}}: {\ begin {cases} u_ {xx} (x, y) u_ {yy} (x, y) = f (x, y) & amp; {\ mbox {in}} \ Omega, \\ u = 0 & amp; {\ mbox {on}} \ partial \ Omega, \ end {cases}}}  ÂÂ$

di mana ${\ displaystyle \ Omega}  ÂÂ$ adalah wilayah terbuka yang terhubung dalam ${\ displaystyle (x, y)}  ÂÂ$ bidang yang batasnya ${\ displaystyle \ partial \ Omega}  ÂÂ$ adalah "bagus" (misalnya, manifold halus atau poligon), dan ${\ displaystyle u_ {xx}}  ÂÂ$ dan ${\ displaystyle u_ {yy}}  ÂÂ$ menunjukkan turunan kedua berkenaan dengan ${\ displaystyle x}  ÂÂ$ dan ${\ displaystyle y}  ÂÂ$ , masing-masing.

P1 problems can be solved "straight" by counting antennivatives. However, this limit value method (BVP) works only when there is one spatial dimension and does not generalize to higher dimension issues or problems such as ${\ displaystyle u u '' = f}$ . For this reason, we will develop a finite element method for P1 and elaborate generalization to P2.

Our explanation will continue in two steps, reflecting two important steps to be taken to solve the boundary value problem (BVP) using FEM.

• In the first step, someone changes the original BVP keyword in its weak form. Little or no calculations are usually required for this step. Transformation is done by hand on paper.
• The second step is discretization, in which the weak form is discretized in a finite-dimensional space.

After this second step, we have a concrete formula for a large linear problem but finite dimensions whose solution will solve the initial BVP problem. This dimension problem is then implemented on the computer.

Weak Formulation

The first step is to convert P1 and P2 to an equivalent weak formulation.

Poor form P1

Jika ${\ displaystyle u}  ÂÂ$ memecahkan P1, lalu untuk semua fungsi halus ${\ displaystyle v}  ÂÂ$ yang memenuhi ketentuan batas perpindahan, yaitu ${\ displaystyle v = 0}  ÂÂ$ di ${\ displaystyle x = 0}  ÂÂ$ dan ${\ displaystyle x = 1}  ÂÂ$ , kami punya

(1) ${\ displaystyle \ int _ {0} ^ {1} f (x) v (x) \, dx = \ int _ {0} ^ {1} u '' (x) v (x) \, dx.}  ÂÂ$

Sebaliknya, jika ${\ displaystyle u}  ÂÂ$ dengan ${\ displaystyle u (0) = u (1) = 0}  ÂÂ$ memenuhi (1) untuk setiap fungsi halus ${\ displaystyle v (x)}  ÂÂ$ maka seseorang dapat menunjukkan bahwa ini ${\ displaystyle u}  ÂÂ$ akan menyelesaikan P1. Buktinya lebih mudah untuk dua kali terus terdiferensiasi ${\ displaystyle u}  ÂÂ$ (teorema nilai rata-rata), tetapi dapat dibuktikan dalam arti distribusi juga.

Kami mendefinisikan fungsi baru ${\ displaystyle \ phi (u, v)}  ÂÂ$ dengan menggunakan integrasi oleh bagian-bagian di sisi kanan (1):

di mana kami telah menggunakan asumsi bahwa ${\ displaystyle v (0) = v (1) = 0}  ÂÂ$ .

Bentuk lemah dari P2

Jika kita mengintegrasikan dengan bagian menggunakan bentuk identitas Green, kita melihat bahwa jika ${\ displaystyle u}  ÂÂ$ menyelesaikan P2, maka kita dapat mendefinisikan ${\ displaystyle \ phi (u, v)}  ÂÂ$ untuk setiap ${\ displaystyle v}  ÂÂ$ oleh

${\ displaystyle \ int _ {\ Omega} fv \, ds = - \ int_ {\ Omega} \ nabla u \ cdot \ nabla v \, ds \ equiv - \ phi (u, v),}  ÂÂ$

di mana ${\ displaystyle \ nabla}  ÂÂ$ menunjukkan gradien dan ${\ displaystyle \ cdot}  ÂÂ$ menunjukkan produk titik dalam bidang dua dimensi. Sekali lagi ${\ displaystyle \, \! \ phi}  ÂÂ$ dapat diubah menjadi produk dalam pada ruang yang sesuai ${\ displaystyle H_ {0} ^ {1} (\ Omega)}  ÂÂ$ dari fungsi "sekali terdiferensiasi" ${\ displaystyle \ Omega}  ÂÂ$ yang nol pada ${\ displaystyle \ partial \ Omega}  ÂÂ$ . Kami juga mengasumsikan bahwa ${\ displaystyle v \ dalam H_ {0} ^ {1} (\ Omega)}  ÂÂ$ (lihat ruang Sobolev). Keberadaan dan keunikan solusi juga dapat ditunjukkan.

Garis besar bukti keberadaan dan keunikan solusi

P1 dan P2 siap untuk didiskritisasi yang mengarah ke sub-masalah umum (3). Ide dasarnya adalah mengganti masalah linear tak hingga dimensi:

Temukan ${\ displaystyle u \ in H_ {0} ^ {1}}  ÂÂ$ seperti itu

${\ displaystyle \ forall v \ in H_ {0} ^ {1}, \; - \ phi (u, v) = \ int fv}  ÂÂ$

dengan versi dimensi-terbatas:

(3) Temukan ${\ displaystyle u \ in V}  ÂÂ$ seperti itu

${\ displaystyle \ forall v \ in V, \; - \ phi (u, v) = \ int fv}  ÂÂ$

di mana ${\ displaystyle V}  ÂÂ$ adalah subruang berdimensi terbatas ${\ displaystyle H_ {0} ^ {1}}  ÂÂ$ . Ada banyak kemungkinan pilihan untuk ${\ displaystyle V}  ÂÂ$ (satu kemungkinan mengarah ke metode spektral). Namun, untuk metode elemen hingga kita mengambil ${\ displaystyle V}  ÂÂ$ menjadi ruang fungsi polinomial piecewise.

Untuk masalah P1

di mana kita mendefinisikan ${\ displaystyle x_ {0} = 0}  ÂÂ$ dan ${\ displaystyle x_ {n 1} = 1}  ÂÂ$ . Amati bahwa fungsi dalam ${\ displaystyle V}  ÂÂ$ tidak terdiferensiasi berdasarkan definisi dasar kalkulus. Memang, jika ${\ displaystyle v \ in V}  ÂÂ$ maka turunannya biasanya tidak didefinisikan pada

Source of the article : Wikipedia