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| :''For the stiffness tensor in solid mechanics, see [[Hooke's law#Matrix representation (stiffness tensor)]].''
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| In the [[finite element method]] for the numerical solution of elliptic [[partial differential equations]], the '''stiffness matrix''' represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.
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| ==The stiffness matrix for the Poisson problem==
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| For simplicity, we will first consider the [[Poisson problem]]
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| :<math> -\nabla^2 u = f</math>
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| on some domain Ω, subject to the boundary condition ''u'' = 0 on the boundary of Ω. To discretize this equation by the finite element method, one chooses a set of ''basis functions'' {''φ''<sub>1</sub>, ..., ''φ''<sub>''n''</sub>} defined on Ω which also vanish on the boundary. One then approximates
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| :<math> u \approx u^h = u_1\varphi_1+\cdots+u_n\varphi_n.</math>
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| The coefficients ''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub> are determined so that the error in the approximation is orthogonal to each basis function ''φ''<sub>''i''</sub>:
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| :<math> \int_\Omega \varphi_i\cdot f \, dx = -\int_\Omega \varphi_i\nabla^2u^h \, dx = -\sum_j\left(\int_\Omega \varphi_i\nabla^2\varphi_j\,dx\right)\, u_j = \sum_j\left(\int_\Omega \nabla\varphi_i\cdot\nabla\varphi_j\, dx\right)u_j.</math>
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| The '''stiffness matrix''' is the n-element square matrix A defined by
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| :<math> A_{ij} = \int_\Omega\nabla\varphi_i\cdot\nabla\varphi_j\, dx.</math>
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| By defining the vector ''F'' with components ''F''<sub>''i''</sub> = (''φ''<sub>''i''</sub>, ''f''), the coefficients ''u''<sub>''i''</sub> are determined by the linear system ''AU'' = ''F''. The stiffness matrix is symmetric, i.e. ''A''<sub>''ij''</sub> = ''A''<sub>''ji''</sub>, so all its eigenvalues are real. Moreover, it is a strictly [[positive-definite matrix]], so that the system ''AU'' = ''F'' always has a unique solution. (For other problems, these nice properties will be lost.)
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| Note that the stiffness matrix will be different depending on the computational grid used for the domain and what type of finite element is used. For example, the stiffness matrix when piecewise quadratic finite elements are used will have more degrees of freedom than piecewise linear elements.
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| ==The stiffness matrix for other problems==
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| Determining the stiffness matrix for other PDE follows essentially the same procedure, but it can be complicated by the choice of boundary conditions. As a more complex example, consider the elliptic equation
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| :<math> -\sum_{k,l}\frac{\partial}{\partial x_k}\left(a^{kl}\frac{\partial u}{\partial x_l}\right) = f</math>
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| where ''A''(''x'') = ''a''<sup>''kl''</sup>(''x'') is a positive-definite matrix defined for each point ''x'' in the domain. We impose the Robin boundary condition
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| :<math> -\sum_{k,l}\nu_k a^{kl}\frac{\partial u}{\partial x_l} = c(u-g),</math>
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| where ''ν''<sub>''k''</sub> is the component of the unit outward normal vector ''ν'' in the ''k''-th direction. The system to be solved is
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| :<math> \sum_j\left(\sum_{k,l}\int_\Omega a^{kl}\frac{\partial\varphi_i}{\partial x_k}\frac{\partial\varphi_j}{\partial x_l}dx+\int_{\partial\Omega}c\varphi_i\varphi_j\, ds\right)u_j = \int_\Omega\varphi_i f\, dx+\int_{\partial\Omega}c\varphi_i g\, ds,</math>
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| as can be shown using an analogue of Green's identity. The coefficients ''u''<sub>''i''</sub> are still found by solving a system of linear equations, but the matrix representing the system is markedly different from that for the ordinary Poisson problem.
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| In general, to each scalar elliptic operator ''L'' of order 2''k'', there is associated a bilinear form ''B'' on the [[Sobolev space]] ''H''<sup>''k''</sup>, so that the [[weak formulation]] of the equation ''Lu'' = ''f'' is
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| :<math> B[u,v] = (f,v)</math>
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| for all functions ''v'' in ''H''<sup>''k''</sup>. Then the stiffness matrix for this problem is
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| :<math> A_{ij} = B[\varphi_j,\varphi_i].</math>
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| ==Practical assembly of the stiffness matrix==
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| In order to implement the finite element method on a computer, one must first choose a set of basis functions and then compute the integrals defining the stiffness matrix. Usually, the domain Ω is discretized by some form of [[mesh generation]], wherein it is divided into non-overlapping triangles or quadrilaterals, which are generally referred to as elements. The basis functions are then chosen to be polynomials of some order within each element, and continuous across element boundaries. The simplest choices are piecewise linear for triangular elements and piecewise bilinear for rectangular elements.
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| The '''element stiffness matrix''' ''A''<sup>[''k'']</sup> for element ''T''<sub>''k''</sub> is the matrix
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| :<math> A^{[k]}_{ij} = \int_{T_k}\nabla\varphi_i\cdot\nabla\varphi_j\, dx.</math>
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| The element stiffness matrix is zero for most values of i and j, for which the corresponding basis functions are zero within ''T''<sub>''k''</sub>. The full stiffness matrix ''A'' is the sum of the element stiffness matrices. In particular, for basis functions that are only supported locally, the stiffness matrix is [[sparse matrix|sparse]].
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| For many standard choices of basis functions, i.e. piecewise linear basis functions on triangles, there are simple formulas for the element stiffness matrices. For example, for piecewise linear elements, consider a triangle with vertices (''x''<sub>1</sub>,''y''<sub>1</sub>), (''x''<sub>2</sub>,''y''<sub>2</sub>), (''x''<sub>3</sub>,''y''<sub>3</sub>), and define the 2×3 matrix
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| :<math> D = \left[\begin{matrix}x_3 - x_2 & x_1 - x_3 & x_2 - x_1 \\ y_3 - y_2 & y_1 - y_3 & y_2 - y_1\end{matrix}\right].</math>
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| Then the element stiffness matrix is
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| :<math> A^{[k]} = D^\top D/4\cdot \mathrm{area}(T)</math>. | |
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| When the differential equation is more complicated, say by having an inhomogeneous diffusion coefficient, the integral defining the element stiffness matrix can be evaluated by [[Gaussian quadrature]].
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| ==References==
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| * {{citation |first1=A. |last1=Ern |first2=J.-L. |last2=Guermond |year=2004 |title=Theory and Practice of Finite Elements |publisher=Springer-Verlag |location=New York, NY |isbn=0387205748 }}
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| * {{citation |first=M.S. |last=Gockenbach |year=2006 |title=Understanding and Implementing the Finite Element Method |publisher=SIAM |location=Philadelphia, PA |isbn=0898716144 }}
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| * {{citation |first1=C. |last1=Grossmann |first2=H.-G. |last2=Roos |first3=M. |last3=Stynes |year=2007 |title=Numerical Treatment of Partial Differential Equations |publisher=Springer-Verlag |location=Berlin, Germany |isbn=978-3-540-71584-9 }}
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| * {{citation |first=C. |last=Johnson |year=2009 |title=Numerical Solution of Partial Differential Equations by the Finite Element Method |publisher=Dover |isbn=978-0486469003 }}
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| * {{citation |first1=O.C. |last1=Zienkiewicz |author1-link=Olgierd Zienkiewicz |first2=R.L. |last2=Taylor |first3=J.Z. |last3=Zhu |year=2005 |title=The Finite Elements Method: Its Basis and Fundamentals |publisher=Elsevier Butterworth-Heinemann |edition=6th |location=Oxford, UK |isbn=978-0750663205 }}
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| [[Category:Applied mathematics]]
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| [[Category:Numerical analysis]]
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They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. Wordpress have every reason with it which promote wordpress development. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Word - Press makes it possible to successfully and manage your website.
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Word - Press installation is very easy and hassle free. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. To do this, you should link your posts to other relevant posts that you've created. There are many advantages of hiring Wordpress developers for Wordpress project development:. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet.
This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. If you adored this write-up and you would certainly such as to receive more information pertaining to wordpress backup plugin kindly go to our own web page. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. with posts or testimonials updated as they are uploaded to a particular section of the website. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts.