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有限元方法基础

The core Partial Differential Equation Toolbox™ algorithm uses the Finite Element Method (FEM) for problems defined on bounded domains in 2-D or 3-D space. In most cases, elementary functions cannot express the solutions of even simple PDEs on complicated geometries. The finite element method describes a complicated geometry as a collection of subdomains by generating a mesh on the geometry. For example, you can approximate the computational domain Ω with a union of triangles (2-D geometry) or tetrahedra (3-D geometry). The subdomains form a mesh, and each vertex is called a node. The next step is to approximate the original PDE problem on each subdomain by using simpler equations.

For example, consider the basic elliptic equation.

$- \nabla \cdot (c \nabla u) + a u = f 上domain ω.$

Suppose that this equation is a subject to the Dirichlet boundary condition $u = r$ 上 $\partial {ω.}_{D}$ 和Neumann boundary conditions on $\partial {ω.}_{N}$ 。这里， $\partial ω. = \partial {ω.}_{D} \cup \partial {ω.}_{N}$ is the boundary of Ω.

The first step in FEM is to convert the original differential (strong) form of the PDE into an integral (弱) form by multiplying with test function $v$ 和integrating over the domain Ω.

$\int_{ω.} (- \nabla \cdot (c \nabla u) + a u - f) v d ω. = 0 \forall v$

测试函数选自在边界的Dirichlet部分上消失的功能集合（功能空间）， $v = 0$ 上 $\partial {ω.}_{D}$ 。以上方程可以你ght of as weighted averaging of the residue using all possible weighting functions $v$ 。允许解决方案的功能集合，金宝搏官方网站u, of the weak form of PDE are chosen so that they satisfy the Dirichlet BC,u=r上 $\partial {ω.}_{D}$ 。

按零件（绿色配方）集成二阶项结果：

$\int_{ω.} (c \nabla u \nabla v + a u v) d ω. - \int_{\partial {ω.}_{N}} \vec{n} \cdot (c \nabla u) v d \partial {ω.}_{N} + \int_{\partial {ω.}_{D}} \vec{n} \cdot (c \nabla u) v d \partial {ω.}_{D} = \int_{ω.} f v d ω. \forall v$

Use the Neumann boundary condition to substitute for second term on the left side of the equation. Also, note that $v = 0$ 上 $\partial {ω.}_{D}$ nullifies the third term. The resulting equation is:

$\int_{ω.} (c \nabla u \nabla v + a u v) d ω. + \int_{\partial {ω.}_{N}} q u v d \partial {ω.}_{N} = \int_{\partial {ω.}_{N}} g v d \partial {ω.}_{N} + \int_{ω.} f v d ω. \forall v$

Note that all manipulations up to this stage are performed on continuum Ω, the global domain of the problem. Therefore, the collection of admissible functions and trial functions span infinite-dimensional functional spaces. Next step is to discretize the weak form by subdividing Ω into smaller subdomains or elements ${ω.}^{e}$ , where $ω. = \cup {ω.}^{e}$ 。该步骤相当于PDE弱形式的投影到有限维子空间上。使用符号 $u_{h}$ 和 $v_{h}$ to represent the finite-dimensional equivalent of admissible and trial functions defined on ${ω.}^{e}$ ，您可以将PDE的离散弱形式写为：

$\int_{{ω.}^{e}} (c \nabla u_{h} \nabla v_{h} + a u_{h} v_{h}) d {ω.}^{e} + \int_{\partial {ω.}_{N}^{e}} q u_{h} v_{h} d \partial {ω.}_{N}^{e} = \int_{\partial {ω.}_{N}^{e}} g v_{h} d \partial {ω.}_{N}^{e} + \int_{{ω.}^{e}} f v_{h} d {ω.}^{e} \forall v_{h}$

接下来，让我们φ._i, withi= 1,2，......，N_p, be the piecewise polynomial basis functions for the subspace containing the collections $u_{h}$ 和 $v_{h}$ 然后是任何特定的 $u_{h}$ 可以表示为基础函数的线性组合：

$u_{h} = \sum_{1}^{N_{p}} U_{i} {φ.}_{i}$

HereU_iare yet undetermined scalar coefficients. Substituting $u_{h}$ 进入离散化的PDE弱形式，并使用每个 $v_{h} = {φ.}_{i}$ as test functions and performing integration over element yields a system ofN_pequations in terms ofN_punknownsU_i。

Note that finite element method approximates a solution by minimizing the associated error function. The minimizing process automatically finds the linear combination of basis functions which is closest to the solutionu。

FEM产生一个系统ku.=Fwhere the matrixK和the right sideFcontain integrals in terms of the test functionsφ._i,φ._j，以及系数c,a,f,q，和g定义问题。解决方案矢量U包含扩展系数u_h, which are also the values ofu_h在每个节点x_k(k= 1,2 for a 2-D problem ork= 1,2,3为3d问题）u_h(x_k) =U_i。

FEM技术也用于解决更多的普遍问题，例如：

Time-dependent problems. The solutionu(x,t) of the equation

$d \frac{\partial u}{\partial t} - \nabla \cdot (c \nabla u) + a u = f$

可以近似

$u_{h} (x, t) = \sum_{i = 1}^{N} U_{i} (t) {φ.}_{i} (x)$

The result is a system of ordinary differential equations (ODEs)

$M \frac{d U}{d t} + K U = F$

Two time derivatives result in a second-order ODE

$M \frac{d^{2} U}{d t^{2}} + K U = F$
Eigenvalue problems. Solve

$- \nabla \cdot (c \nabla u) + a u = λ. d u$

for the unknownsu和λ., whereλ.is a complex number. Using the FEM discretization, you solve the algebraic eigenvalue problemku.=λ.亩找到u_h作为近似u。To solve eigenvalue problems, usesolvepdeeig。
非线性问题。如果是系数c,a,f,q, orgare functions ofuor ∇u, the PDE is called nonlinear and FEM yields a nonlinear systemK(U)U=F(U)。

总结一下，有限元方法：

Represents the original domain of the problem as a collection of elements.
对于每个元素，通过局部近似于原始方程的一组简单方程来替换原始PDE问题。为每个元素的边界应用边界条件。对于系数不依赖于解决方案或其梯度的静止线性问题，结果是方程的线性系统。对于系数取决于解决方案或其梯度的静止问题，结果是非线性方程的系统。对于时间依赖的问题，结果是一组ODES。
将生成的方程和边界条件组装到模拟整个问题的全局方程式系统中。
解决了使用线性溶剂或数值积分的代数方程或杂物的所得系统。工具箱内部呼叫合适的MATLAB^®solvers for this task.

参考资料

[1]库克，罗伯特D.，David S. Malkus和Michael E. Plesha。有限元分析的概念和应用。3rd edition. New York, NY: John Wiley & Sons, 1989.

[2]吉尔伯特斯特朗和乔治修理。有限元法分析。第2版。Wellesley，Ma：Wellesley-Cambridge Press，2008年。

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