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99-450: Altair Radioss is a multidisciplinary finite element solver developed by Altair Engineering . It includes implicit and explicit time integration schemes for the solution of engineering problems, from [[linear] statics and linear dynamics to non-linear transient dynamics and mechanical systems. The multidisciplinary solver has its main strengths in durability, NVH , crash, safety, manufacturability, and fluid-structure interaction. Since

198-1264: A 1 , 1 a 1 , 2 ⋯ a 1 , n a 2 , 1 a 2 , 2 ⋯ a 2 , n ⋮ ⋮ ⋱ ⋮ a n , 1 a n , 2 ⋯ a n , n ] [ v 1 v 2 ⋮ v n ] {\displaystyle {\begin{aligned}A(\mathbf {v} )&=A\left(\sum _{i}v_{i}\mathbf {e} _{i}\right)=\sum _{i}{v_{i}A(\mathbf {e} _{i})}\\&={\begin{bmatrix}A(\mathbf {e} _{1})&A(\mathbf {e} _{2})&\cdots &A(\mathbf {e} _{n})\end{bmatrix}}[\mathbf {v} ]_{E}=A\cdot [\mathbf {v} ]_{E}\\[3pt]&={\begin{bmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\cdots &\mathbf {e} _{n}\end{bmatrix}}{\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}\end{aligned}}} The

297-516: A 2 − 2 a b − 2 a c − 2 a b 1 − 2 b 2 − 2 b c − 2 a c − 2 b c 1 − 2 c 2 ] {\displaystyle \mathbf {A} ={\begin{bmatrix}1-2a^{2}&-2ab&-2ac\\-2ab&1-2b^{2}&-2bc\\-2ac&-2bc&1-2c^{2}\end{bmatrix}}} Note that these are particular cases of

396-1015: A 2 − 2 a b − 2 a c − 2 a d − 2 a b 1 − 2 b 2 − 2 b c − 2 b d − 2 a c − 2 b c 1 − 2 c 2 − 2 c d 0 0 0 1 ] [ x y z 1 ] {\displaystyle {\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}}={\begin{bmatrix}1-2a^{2}&-2ab&-2ac&-2ad\\-2ab&1-2b^{2}&-2bc&-2bd\\-2ac&-2bc&1-2c^{2}&-2cd\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\z\\1\end{bmatrix}}} where d = − p ⋅ N {\displaystyle d=-\mathbf {p} \cdot \mathbf {N} } for some point p {\displaystyle \mathbf {p} } on

495-476: A i , i {\displaystyle a_{i,i}} are zeros leaving only one term in the sum ∑ a i , j e i {\textstyle \sum a_{i,j}\mathbf {e} _{i}} above. The surviving diagonal elements, a i , i {\displaystyle a_{i,i}} , are known as eigenvalues and designated with λ i {\displaystyle \lambda _{i}} in

594-487: A i , j {\displaystyle a_{i,j}} elements of matrix A are determined for a given basis E by applying A to every e j = [ 0 0 ⋯ ( v j = 1 ) ⋯ 0 ] T {\displaystyle \mathbf {e} _{j}={\begin{bmatrix}0&0&\cdots &(v_{j}=1)&\cdots &0\end{bmatrix}}^{\mathrm {T} }} , and observing

693-477: A Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [ x ′ y ′ z ′ 1 ] = [ 1 − 2

792-406: A passive transformation refers to description of the same object as viewed from two different coordinate frames. If one has a linear transformation T ( x ) {\displaystyle T(x)} in functional form, it is easy to determine the transformation matrix A by transforming each of the vectors of the standard basis by T , then inserting the result into the columns of

891-532: A variational formulation , a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation are the Galerkin method , the discontinuous Galerkin method, mixed methods, etc. A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions), and (c)

990-391: A 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear ). More affine transformations can be obtained by composition of two or more affine transformations. For example, given a translation T' with vector ( t x ′ , t y ′ ) , {\displaystyle (t'_{x},t'_{y}),}

1089-409: A common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem: with a finite-dimensional version: where V {\displaystyle V} is a finite-dimensional subspace of H 0 1 {\displaystyle H_{0}^{1}} . There are many possible choices for V {\displaystyle V} (one possibility leads to

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1188-919: A consistent format, suitable for computation. This also allows transformations to be composed easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space R can be represented as linear transformations on the n +1-dimensional space R . These include both affine transformations (such as translation ) and projective transformations . For this reason, 4×4 transformation matrices are widely used in 3D computer graphics . These n +1-dimensional transformation matrices are called, depending on their application, affine transformation matrices , projective transformation matrices , or more generally non-linear transformation matrices . With respect to an n -dimensional matrix, an n +1-dimensional matrix can be described as an augmented matrix . In

1287-407: A continuous domain into a set of discrete sub-domains, usually called elements. Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations that arise from the problem of torsion of a cylinder . Courant's contribution was evolutionary, drawing on

1386-703: A form of Green's identities , we see that if u {\displaystyle u} solves P2, then we may define ϕ ( u , v ) {\displaystyle \phi (u,v)} for any v {\displaystyle v} by ∫ Ω f v d s = − ∫ Ω ∇ u ⋅ ∇ v d s ≡ − ϕ ( u , v ) , {\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),} where ∇ {\displaystyle \nabla } denotes

1485-504: A large body of earlier results for PDEs developed by Lord Rayleigh , Walther Ritz , and Boris Galerkin . The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart , R. W. Clough with co-workers at UC Berkeley , O. C. Zienkiewicz with co-workers Ernest Hinton , Bruce Irons and others at Swansea University , Philippe G. Ciarlet at

1584-494: A larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations . Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis ( FEA ). The subdivision of a whole domain into simpler parts has several advantages: Typical work out of the method involves: The global system of equations has known solution techniques and can be calculated from

1683-802: A line that goes through the origin, let l = ( l x , l y ) {\displaystyle \mathbf {l} =(l_{x},l_{y})} be a vector in the direction of the line. Then use the transformation matrix: A = 1 ‖ l ‖ 2 [ l x 2 − l y 2 2 l x l y 2 l x l y l y 2 − l x 2 ] {\displaystyle \mathbf {A} ={\frac {1}{\lVert \mathbf {l} \rVert ^{2}}}{\begin{bmatrix}l_{x}^{2}-l_{y}^{2}&2l_{x}l_{y}\\2l_{x}l_{y}&l_{y}^{2}-l_{x}^{2}\end{bmatrix}}} To project

1782-453: A matrix. In other words, A = [ T ( e 1 ) T ( e 2 ) ⋯ T ( e n ) ] {\displaystyle A={\begin{bmatrix}T(\mathbf {e} _{1})&T(\mathbf {e} _{2})&\cdots &T(\mathbf {e} _{n})\end{bmatrix}}} For example, the function T ( x ) = 5 x {\displaystyle T(x)=5x}

1881-403: A particular direction by a constant factor but does not affect distances in the perpendicular direction. We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the form x' = kx ; y' = y for some positive constant k . (Note that if k > 1 , then this really is a "stretch"; if k < 1 , it is technically a "compression", but we still call it

1980-402: A particular model class. Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices. Post-processing procedures are designed to extract the data of interest from a finite element solution. To meet

2079-514: A physical system with the underlying physics such as the Euler–Bernoulli beam equation , the heat equation , or the Navier-Stokes equations expressed in either PDE or integral equations , while the divided small elements of the complex problem represent different areas in the physical system. FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines) when

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2178-395: A point through a plane a x + b y + c z = 0 {\displaystyle ax+by+cz=0} (which goes through the origin), one can use A = I − 2 N N T {\displaystyle \mathbf {A} =\mathbf {I} -2\mathbf {NN} ^{\mathrm {T} }} , where I {\displaystyle \mathbf {I} }

2277-1335: A rotation R by an angle θ counter-clockwise , a scaling S with factors ( s x , s y ) {\displaystyle (s_{x},s_{y})} and a translation T of vector ( t x , t y ) , {\displaystyle (t_{x},t_{y}),} the result M of T'RST is: [ s x cos ⁡ θ − s y sin ⁡ θ t x s x cos ⁡ θ − t y s y sin ⁡ θ + t x ′ s x sin ⁡ θ s y cos ⁡ θ t x s x sin ⁡ θ + t y s y cos ⁡ θ + t y ′ 0 0 1 ] {\displaystyle {\begin{bmatrix}s_{x}\cos \theta &-s_{y}\sin \theta &t_{x}s_{x}\cos \theta -t_{y}s_{y}\sin \theta +t'_{x}\\s_{x}\sin \theta &s_{y}\cos \theta &t_{x}s_{x}\sin \theta +t_{y}s_{y}\cos \theta +t'_{y}\\0&0&1\end{bmatrix}}} When using affine transformations,

2376-442: A rotation clockwise (negative direction) about the origin, the functional form is x ′ = x cos ⁡ θ − y sin ⁡ θ {\displaystyle x'=x\cos \theta -y\sin \theta } and y ′ = x sin ⁡ θ + y cos ⁡ θ {\displaystyle y'=x\sin \theta +y\cos \theta }

2475-505: A set of functions of Ω {\displaystyle \Omega } . In the figure on the right, we have illustrated a triangulation of a 15-sided polygonal region Ω {\displaystyle \Omega } in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V {\displaystyle V} would consist of functions that are linear on each triangle of

2574-1140: A special case because they are their own inverses and don't need to be separately calculated. To represent affine transformations with matrices, we can use homogeneous coordinates . This means representing a 2-vector ( x , y ) as a 3-vector ( x , y , 1), and similarly for higher dimensions. Using this system, translation can be expressed with matrix multiplication. The functional form x ′ = x + t x ; y ′ = y + t y {\displaystyle x'=x+t_{x};y'=y+t_{y}} becomes: [ x ′ y ′ 1 ] = [ 1 0 t x 0 1 t y 0 0 1 ] [ x y 1 ] . {\displaystyle {\begin{bmatrix}x'\\y'\\1\end{bmatrix}}={\begin{bmatrix}1&0&t_{x}\\0&1&t_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\1\end{bmatrix}}.} All ordinary linear transformations are included in

2673-440: A stretch. Also, if k = 1 , then the transformation is an identity, i.e. it has no effect.) The matrix associated with a stretch by a factor k along the x-axis is given by: [ k 0 0 1 ] {\displaystyle {\begin{bmatrix}k&0\\0&1\end{bmatrix}}} Similarly, a stretch by a factor k along the y-axis has the form x' = x ; y' = ky , so

2772-728: A vector orthogonally onto a line that goes through the origin, let u = ( u x , u y ) {\displaystyle \mathbf {u} =(u_{x},u_{y})} be a vector in the direction of the line. Then use the transformation matrix: A = 1 ‖ u ‖ 2 [ u x 2 u x u y u x u y u y 2 ] {\displaystyle \mathbf {A} ={\frac {1}{\lVert \mathbf {u} \rVert ^{2}}}{\begin{bmatrix}u_{x}^{2}&u_{x}u_{y}\\u_{x}u_{y}&u_{y}^{2}\end{bmatrix}}} As with reflections,

2871-1515: Is [ x x ( 1 − cos ⁡ θ ) + cos ⁡ θ y x ( 1 − cos ⁡ θ ) − z sin ⁡ θ z x ( 1 − cos ⁡ θ ) + y sin ⁡ θ x y ( 1 − cos ⁡ θ ) + z sin ⁡ θ y y ( 1 − cos ⁡ θ ) + cos ⁡ θ z y ( 1 − cos ⁡ θ ) − x sin ⁡ θ x z ( 1 − cos ⁡ θ ) − y sin ⁡ θ y z ( 1 − cos ⁡ θ ) + x sin ⁡ θ z z ( 1 − cos ⁡ θ ) + cos ⁡ θ ] . {\displaystyle {\begin{bmatrix}xx(1-\cos \theta )+\cos \theta &yx(1-\cos \theta )-z\sin \theta &zx(1-\cos \theta )+y\sin \theta \\xy(1-\cos \theta )+z\sin \theta &yy(1-\cos \theta )+\cos \theta &zy(1-\cos \theta )-x\sin \theta \\xz(1-\cos \theta )-y\sin \theta &yz(1-\cos \theta )+x\sin \theta &zz(1-\cos \theta )+\cos \theta \end{bmatrix}}.} To reflect

2970-1217: Is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j , j ≠ k {\displaystyle x_{j},\;j\neq k} , i.e., v k ( x ) = { x − x k − 1 x k − x k − 1  if  x ∈ [ x k − 1 , x k ] , x k + 1 − x x k + 1 − x k  if  x ∈ [ x k , x k + 1 ] , 0  otherwise , {\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in [x_{k},x_{k+1}],\\0&{\text{ otherwise}},\end{cases}}} for k = 1 , … , n {\displaystyle k=1,\dots ,n} ; this basis

3069-411: Is a connected open region in the ( x , y ) {\displaystyle (x,y)} plane whose boundary ∂ Ω {\displaystyle \partial \Omega } is nice (e.g., a smooth manifold or a polygon ), and u x x {\displaystyle u_{xx}} and u y y {\displaystyle u_{yy}} denote

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3168-597: Is a linear transformation mapping R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} and x {\displaystyle \mathbf {x} } is a column vector with n {\displaystyle n} entries, then T ( x ) = A x {\displaystyle T(\mathbf {x} )=A\mathbf {x} } for some m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called

3267-457: Is a linear transformation. Applying the above process (suppose that n = 2 in this case) reveals that T ( x ) = 5 x = 5 I x = [ 5 0 0 5 ] x {\displaystyle T(\mathbf {x} )=5\mathbf {x} =5I\mathbf {x} ={\begin{bmatrix}5&0\\0&5\end{bmatrix}}\mathbf {x} } The matrix representation of vectors and operators depends on

3366-413: Is a shifted and scaled tent function . For the two-dimensional case, we choose again one basis function v k {\displaystyle v_{k}} per vertex x k {\displaystyle x_{k}} of the triangulation of the planar region Ω {\displaystyle \Omega } . The function v k {\displaystyle v_{k}}

3465-481: Is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations . The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into

3564-725: Is also an inner product, this time on the Lp space L 2 ( 0 , 1 ) {\displaystyle L^{2}(0,1)} . An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique u {\displaystyle u} solving (2) and, therefore, P1. This solution is a-priori only a member of H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} , but using elliptic regularity, will be smooth if f {\displaystyle f} is. P1 and P2 are ready to be discretized, which leads to

3663-403: Is continuous, }}v|_{[x_{k},x_{k+1}]}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}} where we define x 0 = 0 {\displaystyle x_{0}=0} and x n + 1 = 1 {\displaystyle x_{n+1}=1} . Observe that functions in V {\displaystyle V} are not differentiable according to

3762-537: Is easier for twice continuously differentiable u {\displaystyle u} ( mean value theorem ) but may be proved in a distributional sense as well. We define a new operator or map ϕ ( u , v ) {\displaystyle \phi (u,v)} by using integration by parts on the right-hand-side of (1): where we have used the assumption that v ( 0 ) = v ( 1 ) = 0 {\displaystyle v(0)=v(1)=0} . If we integrate by parts using

3861-933: Is given, u {\displaystyle u} is an unknown function of x {\displaystyle x} , and u ″ {\displaystyle u''} is the second derivative of u {\displaystyle u} with respect to x {\displaystyle x} . P2 is a two-dimensional problem ( Dirichlet problem ) P2  : { u x x ( x , y ) + u y y ( x , y ) = f ( x , y )  in  Ω , u = 0  on  ∂ Ω , {\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}} where Ω {\displaystyle \Omega }

3960-427: Is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle). Examples of methods that use higher degree piecewise polynomial basis functions are

4059-518: Is provided only in binary form, and for which third parties are granted "limited permission to use". Finite element The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems ). There are also studies about using FEM solve high-dimensional problems. To solve a problem, the FEM subdivides a large system into smaller, simpler parts called finite elements . This

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4158-789: Is that the inner products ⟨ v j , v k ⟩ = ∫ 0 1 v j v k d x {\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx} and ϕ ( v j , v k ) = ∫ 0 1 v j ′ v k ′ d x {\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx} will be zero for almost all j , k {\displaystyle j,k} . (The matrix containing ⟨ v j , v k ⟩ {\displaystyle \langle v_{j},v_{k}\rangle } in

4257-432: Is that the real plane is mapped to the w = 1 plane in real projective space, and so translation in real Euclidean space can be represented as a shear in real projective space. Although a translation is a non- linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes , in

4356-449: Is the 3×3 identity matrix and N {\displaystyle \mathbf {N} } is the three-dimensional unit vector for the vector normal of the plane. If the L norm of a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} is unity, the transformation matrix can be expressed as: A = [ 1 − 2

4455-407: Is the unique function of V {\displaystyle V} whose value is 1 {\displaystyle 1} at x k {\displaystyle x_{k}} and zero at every x j , j ≠ k {\displaystyle x_{j},\;j\neq k} . Depending on the author, the word "element" in the "finite element method" refers to

4554-883: Is then implemented on a computer . The first step is to convert P1 and P2 into their equivalent weak formulations . If u {\displaystyle u} solves P1, then for any smooth function v {\displaystyle v} that satisfies the displacement boundary conditions, i.e. v = 0 {\displaystyle v=0} at x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} , we have Conversely, if u {\displaystyle u} with u ( 0 ) = u ( 1 ) = 0 {\displaystyle u(0)=u(1)=0} satisfies (1) for every smooth function v ( x ) {\displaystyle v(x)} then one may show that this u {\displaystyle u} will solve P1. The proof

4653-461: Is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all

4752-864: The ( j , k ) {\displaystyle (j,k)} location is known as the Gramian matrix .) In the one dimensional case, the support of v k {\displaystyle v_{k}} is the interval [ x k − 1 , x k + 1 ] {\displaystyle [x_{k-1},x_{k+1}]} . Hence, the integrands of ⟨ v j , v k ⟩ {\displaystyle \langle v_{j},v_{k}\rangle } and ϕ ( v j , v k ) {\displaystyle \phi (v_{j},v_{k})} are identically zero whenever | j − k | > 1 {\displaystyle |j-k|>1} . Similarly, in

4851-483: The Runge-Kutta method . In step (2) above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system . The process is often carried out by FEM software using coordinate data generated from

4950-645: The counter-clockwise rotation matrix from above becomes: [ cos ⁡ θ − sin ⁡ θ 0 sin ⁡ θ cos ⁡ θ 0 0 0 1 ] {\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}} Using transformation matrices containing homogeneous coordinates, translations become linear, and thus can be seamlessly intermixed with all other types of transformations. The reason

5049-841: The gradient and ⋅ {\displaystyle \cdot } denotes the dot product in the two-dimensional plane. Once more ϕ {\displaystyle \,\!\phi } can be turned into an inner product on a suitable space H 0 1 ( Ω ) {\displaystyle H_{0}^{1}(\Omega )} of once differentiable functions of Ω {\displaystyle \Omega } that are zero on ∂ Ω {\displaystyle \partial \Omega } . We have also assumed that v ∈ H 0 1 ( Ω ) {\displaystyle v\in H_{0}^{1}(\Omega )} (see Sobolev spaces ). The existence and uniqueness of

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5148-456: The hp-FEM and spectral FEM . More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques; the most popular are: The primary advantage of this choice of basis

5247-468: The initial values of the original problem to obtain a numerical answer. In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of Galerkin method . The process, in mathematical language,

5346-1163: The matrix multiplication , the homogeneous component w c {\displaystyle w_{c}} will be equal to the value of z {\displaystyle z} and the other three will not change. Therefore, to map back into the real plane we must perform the homogeneous divide or perspective divide by dividing each component by w c {\displaystyle w_{c}} : [ x ′ y ′ z ′ 1 ] = 1 w c [ x c y c z c w c ] = [ x / z y / z 1 1 ] {\displaystyle {\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}}={\frac {1}{w_{c}}}{\begin{bmatrix}x_{c}\\y_{c}\\z_{c}\\w_{c}\end{bmatrix}}={\begin{bmatrix}x/z\\y/z\\1\\1\end{bmatrix}}} More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move

5445-519: The physical sciences , an active transformation is one which actually changes the physical position of a system , and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system ( change of basis ). The distinction between active and passive transformations is important. By default, by transformation , mathematicians usually mean active transformations, while physicists could mean either. Put differently,

5544-1094: The spectral method ). However, we take V {\displaystyle V} as a space of piecewise polynomial functions for the finite element method. We take the interval ( 0 , 1 ) {\displaystyle (0,1)} , choose n {\displaystyle n} values of x {\displaystyle x} with 0 = x 0 < x 1 < ⋯ < x n < x n + 1 = 1 {\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1} and we define V {\displaystyle V} by: V = { v : [ 0 , 1 ] → R : v  is continuous,  v | [ x k , x k + 1 ]  is linear for  k = 0 , … , n , and  v ( 0 ) = v ( 1 ) = 0 } {\displaystyle V=\{v:[0,1]\to \mathbb {R} \;:v{\text{

5643-684: The transformation matrix of T {\displaystyle T} . Note that A {\displaystyle A} has m {\displaystyle m} rows and n {\displaystyle n} columns, whereas the transformation T {\displaystyle T} is from R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in

5742-763: The x axis points right and the y axis points up. For shear mapping (visually similar to slanting), there are two possibilities. A shear parallel to the x axis has x ′ = x + k y {\displaystyle x'=x+ky} and y ′ = y {\displaystyle y'=y} . Written in matrix form, this becomes: [ x ′ y ′ ] = [ 1 k 0 1 ] [ x y ] {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}1&k\\0&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}} A shear parallel to

5841-590: The y axis has x ′ = x {\displaystyle x'=x} and y ′ = y + k x {\displaystyle y'=y+kx} , which has matrix form: [ x ′ y ′ ] = [ 1 0 k 1 ] [ x y ] {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}1&0\\k&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}} For reflection about

5940-498: The 2021 release, Radioss has supported input in the LS-DYNA input format as well as the Radioss 'Block' Format OpenRadioss , a source available software version of Radioss, sharing the capabilities, input and output formats of Altair Radioss, was released on September the 8th 2022. Despite being called open source software , the software cannot be compiled or used without a library that

6039-495: The University of Paris 6 and Richard Gallagher with co-workers at Cornell University . Further impetus was provided in these years by available open-source finite element programs. NASA sponsored the original version of NASTRAN . UC Berkeley made the finite element programs SAP IV and later OpenSees widely available. In Norway, the ship classification society Det Norske Veritas (now DNV GL ) developed Sesam in 1969 for use in

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6138-411: The analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Gilbert Strang and George Fix . The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism , heat transfer , and fluid dynamics . A finite element method is characterized by

6237-1207: The axes is transformed to a rectangle that has the same area as the square. The reciprocal stretch and compression leave the area invariant. For rotation by an angle θ counterclockwise (positive direction) about the origin the functional form is x ′ = x cos ⁡ θ + y sin ⁡ θ {\displaystyle x'=x\cos \theta +y\sin \theta } and y ′ = − x sin ⁡ θ + y cos ⁡ θ {\displaystyle y'=-x\sin \theta +y\cos \theta } . Written in matrix form, this becomes: [ x ′ y ′ ] = [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] [ x y ] {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}} Similarly, for

6336-1323: The chosen basis; a similar matrix will result from an alternate basis. Nevertheless, the method to find the components remains the same. To elaborate, vector v {\displaystyle \mathbf {v} } can be represented in basis vectors, E = [ e 1 e 2 ⋯ e n ] {\displaystyle E={\begin{bmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\cdots &\mathbf {e} _{n}\end{bmatrix}}} with coordinates [ v ] E = [ v 1 v 2 ⋯ v n ] T {\displaystyle [\mathbf {v} ]_{E}={\begin{bmatrix}v_{1}&v_{2}&\cdots &v_{n}\end{bmatrix}}^{\mathrm {T} }} : v = v 1 e 1 + v 2 e 2 + ⋯ + v n e n = ∑ i v i e i = E [ v ] E {\displaystyle \mathbf {v} =v_{1}\mathbf {e} _{1}+v_{2}\mathbf {e} _{2}+\cdots +v_{n}\mathbf {e} _{n}=\sum _{i}v_{i}\mathbf {e} _{i}=E[\mathbf {v} ]_{E}} Now, express

6435-450: The chosen triangulation. One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter h > 0 {\displaystyle h>0} which one takes to be very small. This parameter will be related to

6534-497: The defining equation, which reduces to A e i = λ i e i {\displaystyle A\mathbf {e} _{i}=\lambda _{i}\mathbf {e} _{i}} . The resulting equation is known as eigenvalue equation . The eigenvectors and eigenvalues are derived from it via the characteristic polynomial . With diagonalization , it is often possible to translate to and from eigenbases. Most common geometric transformations that keep

6633-451: The domain changes (as during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations. For example, in a frontal crash simulation, it is possible to increase prediction accuracy in "important" areas like

6732-466: The domain's triangles, the piecewise linear basis function, or both. So, for instance, an author interested in curved domains might replace the triangles with curved primitives and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method

6831-611: The elementary definition of calculus. Indeed, if v ∈ V {\displaystyle v\in V} then the derivative is typically not defined at any x = x k {\displaystyle x=x_{k}} , k = 1 , … , n {\displaystyle k=1,\ldots ,n} . However, the derivative exists at every other value of x {\displaystyle x} , and one can use this derivative for integration by parts . We need V {\displaystyle V} to be

6930-409: The finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM. After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem

7029-464: The finite element method. P1 is a one-dimensional problem  P1  : { u ″ ( x ) = f ( x )  in  ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , {\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}} where f {\displaystyle f}

7128-439: The front of the car and reduce it in its rear (thus reducing the cost of the simulation). Another example would be in numerical weather prediction , where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas. A clear, detailed, and practical presentation of this approach can be found in

7227-432: The homogeneous component of a coordinate vector (normally called w ) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections. Another type of transformation, of importance in 3D computer graphics , is the perspective projection . Whereas parallel projections are used to project points onto the image plane along parallel lines,

7326-401: The largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions V {\displaystyle V} must also change with h {\displaystyle h} . For this reason, one often reads V h {\displaystyle V_{h}} instead of V {\displaystyle V} in

7425-481: The literature. Since we do not perform such an analysis, we will not use this notation. To complete the discretization, we must select a basis of V {\displaystyle V} . In the one-dimensional case, for each control point x k {\displaystyle x_{k}} we will choose the piecewise linear function v k {\displaystyle v_{k}} in V {\displaystyle V} whose value

7524-411: The mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version , hp-version , x-FEM , isogeometric analysis , etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in

7623-534: The matrix associated with this transformation is [ 1 0 0 k ] {\displaystyle {\begin{bmatrix}1&0\\0&k\end{bmatrix}}} If the two stretches above are combined with reciprocal values, then the transformation matrix represents a squeeze mapping : [ k 0 0 1 / k ] . {\displaystyle {\begin{bmatrix}k&0\\0&1/k\end{bmatrix}}.} A square with sides parallel to

7722-573: The matrix form is: [ x ′ y ′ ] = [ cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ] [ x y ] {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}} These formulae assume that

7821-559: The matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A that represents a transformation that "undoes" A since its composition with A is the identity matrix . In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating in opposite direction) and then composing them in reverse order. Reflection matrices are

7920-447: The origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix. A stretch in the xy -plane is a linear transformation which enlarges all distances in

8019-429: The orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be used. The matrix to rotate an angle θ about any axis defined by unit vector ( x , y , z )

8118-407: The perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer (see also reciprocal function ). The simplest perspective projection uses the origin as the center of projection, and

8217-1394: The planar case, if x j {\displaystyle x_{j}} and x k {\displaystyle x_{k}} do not share an edge of the triangulation, then the integrals ∫ Ω v j v k d s {\displaystyle \int _{\Omega }v_{j}v_{k}\,ds} and ∫ Ω ∇ v j ⋅ ∇ v k d s {\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds} are both zero. If we write u ( x ) = ∑ k = 1 n u k v k ( x ) {\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)} and f ( x ) = ∑ k = 1 n f k v k ( x ) {\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)} then problem (3), taking v ( x ) = v j ( x ) {\displaystyle v(x)=v_{j}(x)} for j = 1 , … , n {\displaystyle j=1,\dots ,n} , becomes Transformation matrix In linear algebra , linear transformations can be represented by matrices . If T {\displaystyle T}

8316-1156: The plane at z = 1 {\displaystyle z=1} as the image plane. The functional form of this transformation is then x ′ = x / z {\displaystyle x'=x/z} ; y ′ = y / z {\displaystyle y'=y/z} . We can express this in homogeneous coordinates as: [ x c y c z c w c ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 ] [ x y z 1 ] = [ x y z z ] {\displaystyle {\begin{bmatrix}x_{c}\\y_{c}\\z_{c}\\w_{c}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&1&0\end{bmatrix}}{\begin{bmatrix}x\\y\\z\\1\end{bmatrix}}={\begin{bmatrix}x\\y\\z\\z\end{bmatrix}}} After carrying out

8415-468: The plane, or equivalently, a x + b y + c z + d = 0 {\displaystyle ax+by+cz+d=0} . If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with

8514-471: The requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. Some very efficient postprocessors provide for the realization of superconvergence . The following two problems demonstrate

8613-509: The response vector A e j = a 1 , j e 1 + a 2 , j e 2 + ⋯ + a n , j e n = ∑ i a i , j e i . {\displaystyle A\mathbf {e} _{j}=a_{1,j}\mathbf {e} _{1}+a_{2,j}\mathbf {e} _{2}+\cdots +a_{n,j}\mathbf {e} _{n}=\sum _{i}a_{i,j}\mathbf {e} _{i}.} This equation defines

8712-782: The result of the transformation matrix A upon v {\displaystyle \mathbf {v} } , in the given basis: A ( v ) = A ( ∑ i v i e i ) = ∑ i v i A ( e i ) = [ A ( e 1 ) A ( e 2 ) ⋯ A ( e n ) ] [ v ] E = A ⋅ [ v ] E = [ e 1 e 2 ⋯ e n ] [

8811-532: The right. Since text reads from left to right, column vectors are preferred when transformation matrices are composed: If A and B are the matrices of two linear transformations, then the effect of first applying A and then B to a column vector x {\displaystyle \mathbf {x} } is given by: B ( A x ) = ( B A ) x . {\displaystyle \mathbf {B} (\mathbf {A} \mathbf {x} )=(\mathbf {BA} )\mathbf {x} .} In other words,

8910-406: The same matrix. See homogeneous coordinates and affine transformations below for further explanation. One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed and inverted. Composition is accomplished by matrix multiplication . Row and column vectors are operated upon by matrices, rows on the left and columns on

9009-529: The second derivatives with respect to x {\displaystyle x} and y {\displaystyle y} , respectively. The problem P1 can be solved directly by computing antiderivatives . However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like u + V ″ = f {\displaystyle u+V''=f} . For this reason, we will develop

9108-413: The set of affine transformations, and can be described as a simplified form of affine transformations. Therefore, any linear transformation can also be represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example,

9207-553: The solution can also be shown. We can loosely think of H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} to be the absolutely continuous functions of ( 0 , 1 ) {\displaystyle (0,1)} that are 0 {\displaystyle 0} at x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} (see Sobolev spaces ). Such functions are (weakly) once differentiable, and it turns out that

9306-484: The spatial derivatives from the PDE, thus approximating the PDE locally with These equation sets are element equations. They are linear if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using numerical linear algebra methods. In contrast, ordinary differential equation sets that occur in the transient problems are solved by numerical integration using standard techniques such as Euler's method or

9405-404: The subdomains. The practical application of FEM is known as finite element analysis (FEA). FEA as applied in engineering , is a computational tool for performing engineering analysis . It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually

9504-479: The symmetric bilinear map ϕ {\displaystyle \!\,\phi } then defines an inner product which turns H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side ∫ 0 1 f ( x ) v ( x ) d x {\displaystyle \int _{0}^{1}f(x)v(x)dx}

9603-419: The textbook The Finite Element Method for Engineers . While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering . Its development can be traced back to work by Alexander Hrennikoff and Richard Courant in the early 1940s. Another pioneer

9702-408: The wanted elements, a i , j {\displaystyle a_{i,j}} , of j -th column of the matrix A . Yet, there is a special basis for an operator in which the components form a diagonal matrix and, thus, multiplication complexity reduces to n . Being diagonal means that all coefficients a i , j {\displaystyle a_{i,j}} except

9801-608: Was Ioannis Argyris . In the USSR, the introduction of the practical application of the method is usually connected with the name of Leonard Oganesyan . It was also independently rediscovered in China by Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the finite difference method based on variation principle . Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of

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