When implementing FEM for solving two-dimensional partial differential equations, integrals of the form

Size: px
Start display at page:

Download "When implementing FEM for solving two-dimensional partial differential equations, integrals of the form"

Transcription

1 Quadrature Formulas in Two Dimensions Math 57 - Finite Element Method Section, Spring Shaozhong Deng, PhD (shaodeng@unccedu Dept of Mathematics and Statistics, UNC at Charlotte When implementing FEM for solving two-dimensional partial differential equations, integrals of the form I = F (x, y dxdy have to be evaluated to obtain local stiffness matrices and local load vectors, where usually is an either quadrilateral or triangular element Because the integrand generally depends on user-specified information, such as p(x, y, q(x, y and f(x, y in the second-order self-adjoint elliptic equation: (p(x, y u + q(x, yu = f(x, y, (x, y Ω, in computer programs the analogous integrals shall be evaluated numerically A Recall - Gaussian quadrature in one dimension ( Gaussian quadrature of order N for the standard interval I st = [, ]: g(ξ dξ N w i g(ξ i, ( where ξ i and w i are Gaussian quadrature points and weights Remember that a Gaussian quadrature using N points can provide the exact integral if g(ξ is a polynomial of degree N or less (i Gaussian quadrature of order (one point: g(ξ dξ g( (ii Gaussian quadrature of order (two points: ( g(ξ dξ g ( + g (iii Gaussian quadrature of order (three points: g(ξ dξ 5 ( 9 g 5 ( g ( g 5 ( Gaussian quadrature for general intervals I = [a, b]: Therefore, we have b a x = a + b a x a ( + ξ, or ξ = b a + x b b a F (x dx = b a b a F (x dx b a ( F N w i F a + b a ( + ξ dξ ( a + b a ( + ξ i

2 B Gaussian quadrature for quadrilateral elements ( Gaussian quadrature for the standard quadrilateral element R st = [, ] : (, η (, R st ξ (, (, Figure : The standard quadrilateral element R st I = g(ξ, η dξdη = R st g(ξ, η dξdη For any fixed η, we can integrate numerically with respect to ξ: ( M g(ξ, η dξdη w i g(ξ i, η dη, where ξ i and w i are Gaussian quadrature points and weights of order M in the ξ direction Next integrating numerically with respect to η we have g(ξ, η dξdη M j= N w i ŵ j g(ξ i, η j, where η j and ŵ j are Gaussian quadrature points and weights of order N in the η direction This integration rule is exact if the integrand g(ξ, η contains only the monomials ξ i η j (i =,,, M, j =,,, N Usually M = N (so η i = ξ i, ŵ i = w i, and we have Gaussian quadrature of order N for the standard quadrilateral element g(ξ, η dξdη j= N N w i w j g(ξ i, ξ j ( Example: Gaussian quadrature of order for the standard quadrilateral element R st = [, ] : ( g(ξ, η dξdη g g g, 5 5 (, 5 (, 5 ( g g (, ( g, (, g, 5 5 (, 5 9 g g (, 5 5

3 Figure : Gaussian quadrature points (N = for the standard quadrilateral element R st = [, ] ( Gaussian quadrature for general quadrilateral elements: Let be a quadrilateral element with straight boundary lines and vertices (x i, y i, i =,,, 4 arranged in the counter-clockwise order: y P =(x, y P 4 =(x 4, y 4 x P =(x, y P =(x, y Figure : A quadrilateral element with straight boundary lines We would like to evaluate I = F (x, y dxdy The idea is simple: first transform the quadrilateral element to the standard quadrilateral element R st and then apply the Gaussian quadrature ( How to transform: Using nodal shape functions! Nodal shape functions for quadrilaterals: N (ξ, η = ( ξ( η, 4 (, η (, N (ξ, η = ( + ξ( η, 4 R st N (ξ, η = ( + ξ( + η, 4 ξ N 4 (ξ, η = ( ξ( + η 4 Note N i (ξ, η = at Node i ; at other nodes (, (,

4 Construct a linear mapping to map the quadrilateral element with straight boundary lines to the standard quadrilateral element R st : (x 4, y 4 (x, y y (x, y x (x, y x=p(ξ,η y=q(ξ,η (, R st (, η (, ξ (, Figure 4: Linear mapping between and R st The mapping can be achieved conveniently by using the nodal shape functions as follows: Then we have x = P (ξ, η = y = Q(ξ, η = 4 x i N i (ξ, η = x N (ξ, η + x N (ξ, η + x N (ξ, η + x 4 N 4 (ξ, η, 4 y i N i (ξ, η = y N (ξ, η + y N (ξ, η + y N (ξ, η + y 4 N 4 (ξ, η F (x, y dxdy = F (P (ξ, η, Q(ξ, η J(ξ, η dξdη, R st where J(ξ, η is the Jacobian of the transformation defined by x J(ξ, η = (x, y (ξ, η = ξ x η y ξ y η Applying the Gaussian quadrature ( for the standard quadrilateral element yields Gaussian quadrature of order N for general quadrilateral elements N N F (x, y dxdy w i w j F (P (ξ i, ξ j, Q(ξ i, ξ j J(ξ i, ξ j j= 4

5 C Gaussian quadrature for triangular elements ( Gaussian quadrature for the standard triangular element = {(ξ, η : ξ, η, ξ + η }: (, η (, (, ξ Figure 5: The standard triangular element I = g(ξ, η dξdη = η g(ξ, η dξdη = ξ (a Tensor product-type Gaussian quadrature - Simple but less efficient: g(ξ, η dηdξ Idea: Transform the standard triangular element to the standard quadrilateral element R st, and then apply the Gaussian quadrature for R st Such transformation is defined by ( + a ( b ξ =, 4 η = + b or, a = ξ η, b = η ( b η C a T A B ξ Figure 6: Illustration of the mapping between the square R st and the triangle The transformations defined in ( basically collapse the top edge of the square R st into the top vertex (, of the triangle The Jacobians of the transformations are J(a, b = (ξ, η (a, b = b = η 8 4, J (ξ, η = (a, b (ξ, η = 4 η Therefore, we have I = g(ξ, η dξdη = g(ξ(a, b, η(a, b J(a, b dadb R st 5

6 Remark : Note that one direction of the transformation defined in ( is linear and the corresponding Jacobian is bounded However, the other one is non-linear, and its Jacobian has a singularity Fortunately, this Jacobian is NOT used! Remark : Tensor product-type quadrature have several advantages In particular, their derivation and application is straightforward They are versatile in that many one-dimensional rules are available for several different integrands Extremely high-order polynomials may be evaluated, although precision may be limited since most references provide points and weights to significant digits at most The primary disadvantage is inefficiency since for high N, a relatively large number of points is required, and other quadrature rules are available with many fewer points The secondary disadvantage is that the location of the points is unsymmetrical Except for rules of low order, a large number of points will be concentrated in a relatively small region near one vertex (the top (, vertex Such an arrangement, although correct, maybe considered aesthetically undesirable For details, please read Ref [] (b Symmetrical Gaussian quadrature on triangles: Goal: The goal is to develop quadrature rules of the form g(ξ, η dξdη N g w i g(ξ i, η i, (4 where is the number of quadrature points, (ξ i, η i are quadrature points located inside the standard triangle and w i are weights (normalized with respect to the triangle area In addition to the criteria that the resulting quadrature should use as less as possible number of quadrature points to achieve as high as possible accuracy, we also would like the quadrature points possess some kind of symmetry Gaussian quadrature of degree N for triangles: The basic idea is the same as that used in developing Gaussian quadrature for the standard interval I st = [, ] We want to choose points (ξ i, η i and weights w i in (4 so that the quadrature (4 is as accurate as possible in some sense Generally speaking, a Gaussian quadrature of degree N for triangles is defined as a quadrature of (4 that is exact for arbitrary polynomial of degree N, namely, g(ξ, η dξdη = N g w i g(ξ i, η i, g(ξ, η P N (ξ, η, where P N (ξ, η represents the complete polynomial space of degree N in two dimensions For examples, P N (ξ, η = span { ξ i η j, i, j, i + j N } P (ξ, η = span {, ξ, η}, P (ξ, η = span {, ξ, η, ξ, ξη, η } Note that dim(p N = (N + (N + A useful identity: Therefore, we have ξ i η j dξdη = {, ξ, η, ξ, ξη, η, ξ, ξ η, ξη, η } dξdη = i!j! (i + j +! {, 6, 6,, 4,,, 6, 6, } 6

7 Gaussian quadrature of degree for triangles: By definition, the quadrature should be accurate for g(ξ, η =, ξ, and η We get g(ξ, η = = w i g(ξ, η = ξ 6 = N g w i ξ i g(ξ, η = η 6 = w i η i It is easy to see that =, w = and ξ = η = is a solution Therefore, we have Gaussian quadrature of degree for the standard triangle g(ξ, η dξdη = g Gaussian quadrature of degree for triangles: (, By definition, the quadrature should be accurate for g(ξ, η =, ξ, η, ξ, ξη, and η We get g(ξ, η = = N g w i g(ξ, η = ξ 6 = w i ξ i g(ξ, η = η 6 = w i η i g(ξ, η = ξ = g(ξ, η = ξη 4 = g(ξ, η = η = w i ξi w i ξ i η i w i ηi It is obvious that = will NOT work since we have 6 equations In theory, = may work since in this case we are going to have 6 unknowns (ξ, η, ξ, η, w, w, but the resulting quadrature is NOT symmetric! So we choose = Using = (three points, we have 9 unknowns Because there are only 6 equations, generally speaking the solution is not unique We can verify that (ξ, η = ( 6, 6 is a solution Therefore, we have, (ξ, η = (, 6, (ξ, η = ( 6,, w = w = w = Gaussian quadrature of degree for the standard triangle g(ξ, η dξdη = 6 [ ( g 6, ( + g 6, ( + g 6 6, ] 7

8 Remember that the solution is NOT unique! For example, we can also verify that ( (ξ, η =, ( (, (ξ, η =,, (ξ, η =,, w = w = w = is also a solution Therefore, we have another Gaussian quadrature of degree for the standard triangle g(ξ, η dξdη = 6 Gaussian quadrature of degree for triangles: [ ( g, ( ( + g, + g, ] By definition, the quadrature shall be accurate for g(ξ, η =, ξ, η, ξ, ξη, η, ξ, ξ η, ξη, and η So g(ξ, η = = N g w i g(ξ, η = ξ 6 = w i ξ i g(ξ, η = η 6 = w i η i g(ξ, η = ξ = g(ξ, η = ξη 4 = g(ξ, η = η = g(ξ, η = ξ = g(ξ, η = ξ η 6 = g(ξ, η = ξη 6 = g(ξ, η = η = w i ξi w i ξ i η i w i ηi w i ξi w i ξi η i w i ξ i ηi w i ηi It is obvious that = will NOT work since we have equations So we choose = 4 Using = 4 (four points, we have unknowns Because there are only equations, again generally speaking the solution is not unique We can verify that (ξ, η = (, is a solution Therefore, we have, (ξ, η = ( 5, 5, (ξ, η = w = 7, w = w = w 4 = 5 ( 5, (, (ξ 4, η 4 = 5 5,, 5 8

9 Gaussian quadrature of degree for the standard triangle g(ξ, η dξdη = 7 96 g (, + 5 [ ( g 96 5, ( + g 5 5, ( + g 5 5, ] 5 Again recall that the solution is NOT unique! For example, we can verify that ( (ξ, η =, (, (ξ, η = 5, (, (ξ, η = 5 5, (, (ξ 4, η 4 = 5 5,, 5 w = 7, w = w = w 4 = 5 is also a solution Therefore, we have another Gaussian quadrature of degree for the standard triangle g(ξ, η dξdη = 7 96 g Graphical illustration of quadrature points: (, + 5 [ ( g 96 5, ( + g 5 5, ( + g 5 5, ] 5 * c * b a * a * a * b * c * d (a Linear (b Quadratic (c Cubic a = (, (, w = a = 6, 6, w = a = (,, w = 7 b = (, 6, w = b = ( 5, 5, w = 5 c = ( 6,, w = c = ( 5, 5, w = 5 d = ( 5, 5, w = 5 Figure 7: First set of quadrature rules for triangular elements b * a* c * a a * b * c * d * (a Linear (b Quadratic (c Cubic a = (, (, w = a =,, w = a = (,, w = 7 b = (,, w = b = ( 5, 5, w = 5 c = (,, w = c = ( 5, 5, w = 5 d = ( 5, 5, w = 5 Figure 8: Second set of quadrature rules for triangular elements 9

10 Number of quadrature points : The most commonly referenced Gauss-Legendre locations and weights for triangles are the symmetric quadrature rules of [] This reference provides tables varying from degrees to (79 quadrature points, which are reproduced (for degrees between and in DTriGaussPointsm The weights in these tables are normalized with respect to triangle area, ie, g(ξ, η dξdη N g w i g(ξ i, η i The following tables list the number of quadrature points for degrees to as given in Ref [] It should be mentioned that for some N, the corresponding is not necessarily unique If interested, please see [] and references therein N dim(p N N dim(p N ( Gaussian quadrature for general triangular elements : Let be a triangular element with straight boundary lines and vertices (x i, y i, i =,, arranged in the counter-clockwise order: P =(x, y y P =(x, y x P =(x, y Figure 9: A triangular element with straight boundary lines We would like to evaluate I = F (x, y dxdy, Again the idea is very simple: first transform the triangular element to the standard triangular element and then apply the Gaussian quadrature for as described above How to transform: Using nodal shape functions!

11 Nodal shape functions for triangles: η (, N (ξ, η = ξ η, N (ξ, η = ξ, N (ξ, η = η (, (, ξ Construct a linear mapping to map the general triangular element with straight boundary lines to the standard triangular element : (x, y (x, y y x (x, y x=p(ξ,η y=q(ξ,η η (, (, ξ (, Figure : Linear mapping between and The mapping can be achieved conveniently by using the nodal shape functions as follows: Then we have x = P (ξ, η = y = Q(ξ, η = x i N i (ξ, η = x N (ξ, η + x N (ξ, η + x N (ξ, η, y i N i (ξ, η = y N (ξ, η + y N (ξ, η + y N (ξ, η F (x, y dxdy = F (P (ξ, η, Q(ξ, η J(ξ, η dξdη, where J(ξ, η is the Jacobian of the transformation, namely, x J(ξ, η = (x, y (ξ, η = ξ x η y ξ y = A η Here A represents the area of the triangle, which can be evaluated by Therefore, we have A = x (y y + x (y y + x (y y F (x, y dxdy = A k F (P (ξ, η, Q(ξ, η dξdη Applying the Gaussian quadrature of degree N for the standard triangular element (4 yields

12 Gaussian quadrature of degree N for general triangular elements N g F (x, y dxdy A w i F (P (ξ i, η i, Q(ξ i, η i (5 Note that the inverse of the transformation shown in Fig is NOT needed, but it can be found that ξ = A [(y y (x x (x x (y y ], η = A [ (y y (x x + (x x (y y ] ( Implementation: The Matlab code using the Gaussian quadrature (5 to evaluate I = F (x, y dxdy is int fm It uses fm which defines the function F (x, y and TriGaussPointsm which provides Gaussian points and weights The calling sequence of int fm is int_f(quadrature_degree, x, x, x, y, y, y where (x, y, (x, y, and (x, y are the coordinates of the three vertices of the triangular element A simple test: Consider the following integral I = ( x y dxdy =, where is the triangle defined by the three vertices: (,, (, / and (, Since f(x, y = x y has degree of, Gaussian quadrature with N = should be able to provide the exact integral Actually if we run int_f(,,,,, 5, in Matlab, it returns, which is / D Contour integrals In many cases (such as when natural or mixed boundary conditions are involved we have to evaluate integrals along the boundary of a (boundary element which are of the form: I = Pj P i B(x, y ds, where P i and P j are consecutive nodal points in the counter-clockwise order and ds = dx + dy is the differential arc length along the boundary of the element The idea is: first transform the straight contour P i P j to an interval I = [a, b], and then apply the Gaussian quadrature for the interval

13 y P j =(x j, y j x P i =(x i, y i Figure : A (straight contour Suppose that the following transformation is used to transform a general quadrilateral/triangular element to the standard quadrilateral/triangular element R st or : x = P (ξ, η, y = Q(ξ, η Then, we have dx = x x dξ + ξ η dη = J dξ + J dη, dy = y y dξ + ξ η dη = J dξ + J dη, (6a (6b where J, J, etc, are elements of the Jacobian matrix J(ξ, η The contour P i P j is a boundary line of a quadrilateral element: (x 4, y 4 (x, y y (x, y x (x, y x=p(ξ,η y=q(ξ,η (, R st (, η (, ξ (, Along each side of a quadrilateral element, either ξ or η is fixed For example, along side (P P, η = so dη = In this case, we have ( x dx = dξ = J (ξ, dξ, ξ η= ( y dy = dξ = J (ξ, dξ ξ η= Therefore, ds = J (ξ, + J (ξ, dξ,

14 and accordingly, we have Similarly, we have P P P B(x, y ds = P B(x, y ds = P4 P P P 4 B(x, y ds = B(x, y ds = B(P (ξ,, Q(ξ, J (ξ, + J (ξ, dξ B(P (, η, Q(, η J (, η + J (, η dη, B(P (ξ,, Q(ξ, J (ξ, + J (ξ, dξ, B(P (, η, Q(, η J (, η + J (, η dη The contour P i P j is a boundary line of a triangular element: (x, y (x, y y x (x, y x=p(ξ,η y=q(ξ,η η (, (, ξ (, Along side (P P or (P P of a triangular element, either ξ or η is fixed Similarly, we have P P B(x, y ds = P P B(x, y ds = B(P (ξ,, Q(ξ, J (ξ, + J (ξ, dξ, B(P (, η, Q(, η J (, η + J (, η dη On the other hand, along side (P P, ξ + η = so dξ = dη In this case, we have dx = J dξ + J dη = [J ( η, η J ( η, η] dη, dy = J dξ + J dη = [J ( η, η J ( η, η] dη Therefore, ds = [J ( η, η J ( η, η] + [J ( η, η J ( η, η] dη = H(ηdη, and accordingly, we have P P B(x, y ds = B(P ( η, η, Q( η, ηh(η dη 4

15 Appendix A - Simplex coordinates The Gaussian quadrature for triangles described above can be applied to integrals over any triangular domain T in either two or three dimensions: I = F (r dr T In three dimensions, however, it is more convenient to consider these integrals in terms of simplex coordinates, also called area coordinates or barycentric coordinates To develop the simplex coordinate transformation, consider the triangle T defined by the vertices v = (x, y, z, v = (x, y, z, and v = (x, y, z and the edges e = v v, e = v v, and e = v v Any point r located on the triangle can be written as a weighted sum of these three nodes via where α, β, and γ are the simplex coordinates defined by r = γv + αv + βv, (7 α = A A, β = A A, γ = A A, and A is the area of T These coordinates are subject to the constraint Therefore, and α + β + γ = γ = α β r = ( α βv + αv + βv Note that (7 has the form of a linear interpolation Indeed, simplex coordinates will also allow us to perform a linear interpolation of a function F (x, y at points inside the triangle if its values are known at the vertices v (, β A v A A z y x v (, (, α (a The original triangle (b The transformed triangle Figure : Simplex coordinates for a triangle The resulting transformation of the integral to simplex coordinates is F (r dr = F (α, β J(α, β dαdβ T Again, it can be proved that the Jacobian is J(α, β = A, where A is the area of the triangle T, which can be evaluated by A = (v v (v v 5

16 Finally, we have T F (r dr = A F (α, β dαdβ = A α F (α, β dβdα From (x, y, z to simplex coordinates: Given a point r = (x, y, z inside a triangle, sometimes it is also desirable to obtain the simplex coordinates (α, β, γ We can write the barycentric expansion of r = (x, y, z in terms of the components of the triangle vertices as Substituting γ = α β into the above gives Rearranging, these are x = γx + αx + βx, y = γy + αy + βy, z = γz + αz + βz x = ( α βx + αx + βx, y = ( α βy + αy + βy, z = ( α βz + αz + βz α(x x + β(x x + x x =, α(y y + β(y y + y y =, α(z z + β(z z + z z = Solving for α and β gives us and where α = β = B(F + I C(E + H A(E + H B(D + G A(F + I C(D + G B(D + G A(E + H, A = x x, B = x x, C = x x, D = y y, E = y y, F = y y, G = z z, H = z z, I = z z 6

17 Appendix B - Matlab code of Gaussian quadrature for triangles ( Main routine: int fm function z = int_f(n,x,x,x,y,y,y % This function evaluates \iint_ f(x,y dxdy using % the Gaussian quadrature of order N where is a % triangle with vertices (x,y, (x,y and (x,y xw = TriGaussPoints(N; % get quadrature points and weights % calculate the area of the triangle A=abs(x*(y-y+x*(y-y+x*(y-y/ % find number of Gauss points NP=length(xw(:,; z = ; for j = :NP x = x*(-xw(j,-xw(j,+x*xw(j,+x*xw(j, y = y*(-xw(j,-xw(j,+y*xw(j,+y*xw(j, z = z + f(x,y*xw(j,; end z = A*z; return end ( Function to define F (x, y: fm function z=f(x,y z=-x-*y; return; end ( Function to provide Gaussian points (ξ i, η i and weights w i : TriGaussPointsm function xw = TriGaussPoints(n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function TriGaussPoints provides the Gaussian points and weights % % for the Gaussian quadrature of order n for the standard triangles % % % % Input: n - the order of the Gaussian quadrature (n<= % % % % Output: xw - a n by matrix: % % st column gives the x-coordinates of points % % nd column gives the y-coordinates of points % % rd column gives the weights % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7

18 xw = zeros(n,; if (n == xw=[ ]; elseif (n == xw=[ ]; elseif (n == xw=[ ]; elseif (n == 4 xw=[ ]; elseif (n == 5 xw=[ ]; elseif (n == 6 xw=[ ]; elseif (n == 7 xw=[

19 ]; elseif (n == 8 xw=[ ]; (cases for n>=9 omitted in the print-out else error( Bad input n ; end return end References [] D A Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int J Num Meth Engng, (985:9- [] B Szabó, I Babu ska, Finite Element Analysis, Wiley, New York, 99 9

AMS527: Numerical Analysis II

AMS527: Numerical Analysis II AMS527: Numerical Analysis II A Brief Overview of Finite Element Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao SUNY Stony Brook AMS527: Numerical Analysis II 1 / 25 Overview Basic concepts Mathematical

More information

Appropriate Gaussian quadrature formulae for triangles

Appropriate Gaussian quadrature formulae for triangles International Journal of Applied Mathematics and Computation Journal homepage: www.darbose.in/ijamc ISSN: 097-665 (Print) 097-673 (Online) Volume (1) 01 3 Appropriate Gaussian quadrature formulae for triangles

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 36 In last class, we have derived element equations for two d elasticity problems

More information

Set No. 1 IV B.Tech. I Semester Regular Examinations, November 2010 FINITE ELEMENT METHODS (Mechanical Engineering) Time: 3 Hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks

More information

Numerical Integration

Numerical Integration Numerical Integration Numerical Integration is the process of computing the value of a definite integral, when the values of the integrand function, are given at some tabular points. As in the case of

More information

Shape Functions, Derivatives, and Integration

Shape Functions, Derivatives, and Integration Shape Functions, Derivatives, and Integration CHAPTER 6 6. Introduction In the previous chapter we found that the quasi-harmonic equation created a weak form that contained derivatives of the dependent

More information

Finite Element Methods

Finite Element Methods Chapter 5 Finite Element Methods 5.1 Finite Element Spaces Remark 5.1 Mesh cells, faces, edges, vertices. A mesh cell is a compact polyhedron in R d, d {2,3}, whose interior is not empty. The boundary

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 24

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 24 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 24 So in today s class, we will look at quadrilateral elements; and we will

More information

Edge and local feature detection - 2. Importance of edge detection in computer vision

Edge and local feature detection - 2. Importance of edge detection in computer vision Edge and local feature detection Gradient based edge detection Edge detection by function fitting Second derivative edge detectors Edge linking and the construction of the chain graph Edge and local feature

More information

CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE

CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE CHAPTER 5 NUMERICAL INTEGRATION METHODS OVER N- DIMENSIONAL REGIONS USING GENERALIZED GAUSSIAN QUADRATURE 5.1 Introduction Multidimensional integration appears in many mathematical models and can seldom

More information

Integration of Singular Enrichment Functions in the Generalized/Extended Finite Element Method for Three-Dimensional Problems

Integration of Singular Enrichment Functions in the Generalized/Extended Finite Element Method for Three-Dimensional Problems Submitted to International Journal for Numerical Methods in Engineering Integration of Singular Enrichment Functions in the Generalized/Extended Finite Element Method for Three-Dimensional Problems Kyoungsoo

More information

Documentation for Numerical Derivative on Discontinuous Galerkin Space

Documentation for Numerical Derivative on Discontinuous Galerkin Space Documentation for Numerical Derivative on Discontinuous Galerkin Space Stefan Schnake 204 Introduction This documentation gives a guide to the syntax and usage of the functions in this package as simply

More information

SYSTEMS OF NONLINEAR EQUATIONS

SYSTEMS OF NONLINEAR EQUATIONS SYSTEMS OF NONLINEAR EQUATIONS Widely used in the mathematical modeling of real world phenomena. We introduce some numerical methods for their solution. For better intuition, we examine systems of two

More information

MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA

MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA A. N. Johnson et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 3, No. 3 (2015) 269 278 MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA

More information

CS 450 Numerical Analysis. Chapter 7: Interpolation

CS 450 Numerical Analysis. Chapter 7: Interpolation Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80

More information

Finite Element Method. Chapter 7. Practical considerations in FEM modeling

Finite Element Method. Chapter 7. Practical considerations in FEM modeling Finite Element Method Chapter 7 Practical considerations in FEM modeling Finite Element Modeling General Consideration The following are some of the difficult tasks (or decisions) that face the engineer

More information

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons Noname manuscript No. (will be inserted by the editor) Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons S. E. Mousavi N. Sukumar Received: date

More information

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons Comput Mech (2011) 47:535 554 DOI 10.1007/s00466-010-0562-5 ORIGINAL PAPER Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons S. E. Mousavi N.

More information

AMSC/CMSC 460 Final Exam, Fall 2007

AMSC/CMSC 460 Final Exam, Fall 2007 AMSC/CMSC 460 Final Exam, Fall 2007 Show all work. You may leave arithmetic expressions in any form that a calculator could evaluate. By putting your name on this paper, you agree to abide by the university

More information

On the maximum rank of completions of entry pattern matrices

On the maximum rank of completions of entry pattern matrices Linear Algebra and its Applications 525 (2017) 1 19 Contents lists available at ScienceDirect Linear Algebra and its Applications wwwelseviercom/locate/laa On the maximum rank of completions of entry pattern

More information

Computational Physics PHYS 420

Computational Physics PHYS 420 Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt

More information

7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

Chapter 7 Practical Considerations in Modeling. Chapter 7 Practical Considerations in Modeling

Chapter 7 Practical Considerations in Modeling. Chapter 7 Practical Considerations in Modeling CIVL 7/8117 1/43 Chapter 7 Learning Objectives To present concepts that should be considered when modeling for a situation by the finite element method, such as aspect ratio, symmetry, natural subdivisions,

More information

Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems

Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 78:1220 1257 Published online 23 December 2008 in Wiley InterScience (www.interscience.wiley.com)..2530 Integration

More information

A MIXED QUADRATURE FORMULA USING RULES OF LOWER ORDER

A MIXED QUADRATURE FORMULA USING RULES OF LOWER ORDER Bulletin of the Marathwada Mathematical Society Vol.5, No., June 004, Pages 6-4 ABSTRACT A MIXED QUADRATURE FORMULA USING RULES OF LOWER ORDER Namita Das And Sudhir Kumar Pradhan P.G. Department of Mathematics

More information

An introduction to interpolation and splines

An introduction to interpolation and splines An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve

More information

DISCRETE MATHEMATICAL FILTERS OF VIDEO INFORMATION PROCESSING FOR AUTOMATIC IMAGE COMPONENTS DETECTION AND MARKING

DISCRETE MATHEMATICAL FILTERS OF VIDEO INFORMATION PROCESSING FOR AUTOMATIC IMAGE COMPONENTS DETECTION AND MARKING DISCRETE MATHEMATICAL FILTERS OF VIDEO INFORMATION PROCESSING FOR AUTOMATIC IMAGE COMPONENTS DETECTION AND MARKING OVIDIU ILIE ŞANDRU 1, LUIGE VLĂDĂREANU, ALEXANDRA ŞANDRU 3 1 Abstract. In this paper we

More information

An introduction to mesh generation Part IV : elliptic meshing

An introduction to mesh generation Part IV : elliptic meshing Elliptic An introduction to mesh generation Part IV : elliptic meshing Department of Civil Engineering, Université catholique de Louvain, Belgium Elliptic Curvilinear Meshes Basic concept A curvilinear

More information

Polynomials tend to oscillate (wiggle) a lot, even when our true function does not.

Polynomials tend to oscillate (wiggle) a lot, even when our true function does not. AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 2: Spline Approximations Dianne P O Leary c 2001, 2002, 2007 Piecewise polynomial interpolation Piecewise polynomial interpolation Read: Chapter 3 Skip:

More information

Element Quality Metrics for Higher-Order Bernstein Bézier Elements

Element Quality Metrics for Higher-Order Bernstein Bézier Elements Element Quality Metrics for Higher-Order Bernstein Bézier Elements Luke Engvall and John A. Evans Abstract In this note, we review the interpolation theory for curvilinear finite elements originally derived

More information

An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001)

An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001) An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (000/001) Summary The objectives of this project were as follows: 1) Investigate iterative

More information

Nodal Basis Functions for Serendipity Finite Elements

Nodal Basis Functions for Serendipity Finite Elements Nodal Basis Functions for Serendipity Finite Elements Andrew Gillette Department of Mathematics University of Arizona joint work with Michael Floater (University of Oslo) Andrew Gillette - U. Arizona Nodal

More information

MATH3016: OPTIMIZATION

MATH3016: OPTIMIZATION MATH3016: OPTIMIZATION Lecturer: Dr Huifu Xu School of Mathematics University of Southampton Highfield SO17 1BJ Southampton Email: h.xu@soton.ac.uk 1 Introduction What is optimization? Optimization is

More information

A new 8-node quadrilateral spline finite element

A new 8-node quadrilateral spline finite element Journal of Computational and Applied Mathematics 195 (2006) 54 65 www.elsevier.com/locate/cam A new 8-node quadrilateral spline finite element Chong-Jun Li, Ren-Hong Wang Institute of Mathematical Sciences,

More information

Algorithms and Data Structures

Algorithms and Data Structures Charles A. Wuethrich Bauhaus-University Weimar - CogVis/MMC June 22, 2017 1/51 Introduction Matrix based Transitive hull All shortest paths Gaussian elimination Random numbers Interpolation and Approximation

More information

Parameterization. Michael S. Floater. November 10, 2011

Parameterization. Michael S. Floater. November 10, 2011 Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point

More information

Constrained and Unconstrained Optimization

Constrained and Unconstrained Optimization Constrained and Unconstrained Optimization Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Oct 10th, 2017 C. Hurtado (UIUC - Economics) Numerical

More information

Math 5593 Linear Programming Lecture Notes

Math 5593 Linear Programming Lecture Notes Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................

More information

38. Triple Integration over Rectangular Regions

38. Triple Integration over Rectangular Regions 8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.

More information

Math 226A Homework 4 Due Monday, December 11th

Math 226A Homework 4 Due Monday, December 11th Math 226A Homework 4 Due Monday, December 11th 1. (a) Show that the polynomial 2 n (T n+1 (x) T n 1 (x)), is the unique monic polynomial of degree n + 1 with roots at the Chebyshev points x k = cos ( )

More information

f xx + f yy = F (x, y)

f xx + f yy = F (x, y) Application of the 2D finite element method to Laplace (Poisson) equation; f xx + f yy = F (x, y) M. R. Hadizadeh Computer Club, Department of Physics and Astronomy, Ohio University 4 Nov. 2013 Domain

More information

arxiv: v1 [math.na] 20 Sep 2016

arxiv: v1 [math.na] 20 Sep 2016 arxiv:1609.06236v1 [math.na] 20 Sep 2016 A Local Mesh Modification Strategy for Interface Problems with Application to Shape and Topology Optimization P. Gangl 1,2 and U. Langer 3 1 Doctoral Program Comp.

More information

Bernstein-Bezier Splines on the Unit Sphere. Victoria Baramidze. Department of Mathematics. Western Illinois University

Bernstein-Bezier Splines on the Unit Sphere. Victoria Baramidze. Department of Mathematics. Western Illinois University Bernstein-Bezier Splines on the Unit Sphere Victoria Baramidze Department of Mathematics Western Illinois University ABSTRACT I will introduce scattered data fitting problems on the sphere and discuss

More information

Adaptive numerical methods

Adaptive numerical methods METRO MEtallurgical TRaining On-line Adaptive numerical methods Arkadiusz Nagórka CzUT Education and Culture Introduction Common steps of finite element computations consists of preprocessing - definition

More information

Level-set and ALE Based Topology Optimization Using Nonlinear Programming

Level-set and ALE Based Topology Optimization Using Nonlinear Programming 10 th World Congress on Structural and Multidisciplinary Optimization May 19-24, 2013, Orlando, Florida, USA Level-set and ALE Based Topology Optimization Using Nonlinear Programming Shintaro Yamasaki

More information

Collocation and optimization initialization

Collocation and optimization initialization Boundary Elements and Other Mesh Reduction Methods XXXVII 55 Collocation and optimization initialization E. J. Kansa 1 & L. Ling 2 1 Convergent Solutions, USA 2 Hong Kong Baptist University, Hong Kong

More information

) 2 + (y 2. x 1. y c x2 = y

) 2 + (y 2. x 1. y c x2 = y Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since this

More information

A popular method for moving beyond linearity. 2. Basis expansion and regularization 1. Examples of transformations. Piecewise-polynomials and splines

A popular method for moving beyond linearity. 2. Basis expansion and regularization 1. Examples of transformations. Piecewise-polynomials and splines A popular method for moving beyond linearity 2. Basis expansion and regularization 1 Idea: Augment the vector inputs x with additional variables which are transformation of x use linear models in this

More information

= f (a, b) + (hf x + kf y ) (a,b) +

= f (a, b) + (hf x + kf y ) (a,b) + Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis

3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis Adv. Theor. Appl. Mech., Vol. 1, 2008, no. 3, 131-139 3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis Yunhua Luo Department of Mechanical & Manufacturing Engineering, University of

More information

Module: 2 Finite Element Formulation Techniques Lecture 3: Finite Element Method: Displacement Approach

Module: 2 Finite Element Formulation Techniques Lecture 3: Finite Element Method: Displacement Approach 11 Module: 2 Finite Element Formulation Techniques Lecture 3: Finite Element Method: Displacement Approach 2.3.1 Choice of Displacement Function Displacement function is the beginning point for the structural

More information

Chapter 6. Curves and Surfaces. 6.1 Graphs as Surfaces

Chapter 6. Curves and Surfaces. 6.1 Graphs as Surfaces Chapter 6 Curves and Surfaces In Chapter 2 a plane is defined as the zero set of a linear function in R 3. It is expected a surface is the zero set of a differentiable function in R n. To motivate, graphs

More information

Lecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10

Lecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful

More information

Background for Surface Integration

Background for Surface Integration Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

More information

Parameterization of triangular meshes

Parameterization of triangular meshes Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to

More information

Interpolation and Splines

Interpolation and Splines Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values

More information

THREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.

THREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions. THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of

More information

FRUCHT S THEOREM FOR THE DIGRAPH FACTORIAL

FRUCHT S THEOREM FOR THE DIGRAPH FACTORIAL Discussiones Mathematicae Graph Theory 33 (2013) 329 336 doi:10.7151/dmgt.1661 FRUCHT S THEOREM FOR THE DIGRAPH FACTORIAL Richard H. Hammack Department of Mathematics and Applied Mathematics Virginia Commonwealth

More information

February 23 Math 2335 sec 51 Spring 2016

February 23 Math 2335 sec 51 Spring 2016 February 23 Math 2335 sec 51 Spring 2016 Section 4.1: Polynomial Interpolation Interpolation is the process of finding a curve or evaluating a function whose curve passes through a known set of points.

More information

Numerical Methods for PDEs : Video 11: 1D FiniteFebruary Difference 15, Mappings Theory / 15

Numerical Methods for PDEs : Video 11: 1D FiniteFebruary Difference 15, Mappings Theory / 15 22.520 Numerical Methods for PDEs : Video 11: 1D Finite Difference Mappings Theory and Matlab February 15, 2015 22.520 Numerical Methods for PDEs : Video 11: 1D FiniteFebruary Difference 15, Mappings 2015

More information

2.2 Transformers: More Than Meets the y s

2.2 Transformers: More Than Meets the y s 10 SECONDARY MATH II // MODULE 2 STRUCTURES OF EXPRESSIONS 2.2 Transformers: More Than Meets the y s A Solidify Understanding Task Writetheequationforeachproblembelow.Useasecond representationtocheckyourequation.

More information

Separation of variables: Cartesian coordinates

Separation of variables: Cartesian coordinates Separation of variables: Cartesian coordinates October 3, 15 1 Separation of variables in Cartesian coordinates The separation of variables technique is more powerful than the methods we have studied so

More information

PROPERTIES OF NATURAL ELEMENT COORDINATES ON ANY POLYHEDRON

PROPERTIES OF NATURAL ELEMENT COORDINATES ON ANY POLYHEDRON PROPRTIS OF NATURAL LMNT COORDINATS ON ANY POLYHDRON P. Milbradt and T. Fröbel Institute of Computer Science in Civil ngineering, Univercity of Hanover, 3067, Hanover, Germany; PH (+49) 5-76-5757; FAX

More information

IN FINITE element simulations usually two computation

IN FINITE element simulations usually two computation Proceedings of the International Multiconference on Computer Science and Information Technology pp. 337 342 Higher order FEM numerical integration on GPUs with OpenCL 1 Przemysław Płaszewski, 12 Krzysztof

More information

MATH 890 HOMEWORK 2 DAVID MEREDITH

MATH 890 HOMEWORK 2 DAVID MEREDITH MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet

More information

4. Simplicial Complexes and Simplicial Homology

4. Simplicial Complexes and Simplicial Homology MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 4. Simplicial Complexes and Simplicial Homology Geometric simplicial complexes 4.1 Definition. A finite subset { v 0, v 1,..., v r } R n

More information

Computational Fluid Dynamics - Incompressible Flows

Computational Fluid Dynamics - Incompressible Flows Computational Fluid Dynamics - Incompressible Flows March 25, 2008 Incompressible Flows Basis Functions Discrete Equations CFD - Incompressible Flows CFD is a Huge field Numerical Techniques for solving

More information

Linear Finite Element Methods

Linear Finite Element Methods Chapter 3 Linear Finite Element Methods The finite element methods provide spaces V n of functions that are piecewise smooth and simple, and locally supported basis function of these spaces to achieve

More information

Open and Closed Sets

Open and Closed Sets Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

More information

1.2 Numerical Solutions of Flow Problems

1.2 Numerical Solutions of Flow Problems 1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian

More information

Isogeometric Collocation Method

Isogeometric Collocation Method Chair for Computational Analysis of Technical Systems Faculty of Mechanical Engineering, RWTH Aachen University Isogeometric Collocation Method Seminararbeit By Marko Blatzheim Supervisors: Dr. Stefanie

More information

Interpolation by Spline Functions

Interpolation by Spline Functions Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves

More information

MA 323 Geometric Modelling Course Notes: Day 14 Properties of Bezier Curves

MA 323 Geometric Modelling Course Notes: Day 14 Properties of Bezier Curves MA 323 Geometric Modelling Course Notes: Day 14 Properties of Bezier Curves David L. Finn In this section, we discuss the geometric properties of Bezier curves. These properties are either implied directly

More information

The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a

The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with

More information

Math 113 Exam 1 Practice

Math 113 Exam 1 Practice Math Exam Practice January 6, 00 Exam will cover sections 6.-6.5 and 7.-7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second

More information

Contents. I The Basic Framework for Stationary Problems 1

Contents. I The Basic Framework for Stationary Problems 1 page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other

More information

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to

More information

Introduction to Computer Graphics. Modeling (3) April 27, 2017 Kenshi Takayama

Introduction to Computer Graphics. Modeling (3) April 27, 2017 Kenshi Takayama Introduction to Computer Graphics Modeling (3) April 27, 2017 Kenshi Takayama Solid modeling 2 Solid models Thin shapes represented by single polygons Unorientable Clear definition of inside & outside

More information

Numerische Mathematik

Numerische Mathematik Numer. Math. (00) 98: 559 579 Digital Object Identifier (DOI) 10.1007/s0011-00-0536-7 Numerische Mathematik On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation

More information

Generalized barycentric coordinates

Generalized barycentric coordinates Generalized barycentric coordinates Michael S. Floater August 20, 2012 In this lecture, we review the definitions and properties of barycentric coordinates on triangles, and study generalizations to convex,

More information

Inverse and Implicit functions

Inverse and Implicit functions CHAPTER 3 Inverse and Implicit functions. Inverse Functions and Coordinate Changes Let U R d be a domain. Theorem. (Inverse function theorem). If ϕ : U R d is differentiable at a and Dϕ a is invertible,

More information

Parametric Surfaces. Substitution

Parametric Surfaces. Substitution Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional

More information

A GENTLE INTRODUCTION TO THE BASIC CONCEPTS OF SHAPE SPACE AND SHAPE STATISTICS

A GENTLE INTRODUCTION TO THE BASIC CONCEPTS OF SHAPE SPACE AND SHAPE STATISTICS A GENTLE INTRODUCTION TO THE BASIC CONCEPTS OF SHAPE SPACE AND SHAPE STATISTICS HEMANT D. TAGARE. Introduction. Shape is a prominent visual feature in many images. Unfortunately, the mathematical theory

More information

Integration. Volume Estimation

Integration. Volume Estimation Monte Carlo Integration Lab Objective: Many important integrals cannot be evaluated symbolically because the integrand has no antiderivative. Traditional numerical integration techniques like Newton-Cotes

More information

Surface Parameterization

Surface Parameterization Surface Parameterization A Tutorial and Survey Michael Floater and Kai Hormann Presented by Afra Zomorodian CS 468 10/19/5 1 Problem 1-1 mapping from domain to surface Original application: Texture mapping

More information

Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 3, 2017, Lesson 1

Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 3, 2017, Lesson 1 Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 3, 2017, Lesson 1 1 Politecnico di Milano, February 3, 2017, Lesson 1 2 Outline

More information

On a nested refinement of anisotropic tetrahedral grids under Hessian metrics

On a nested refinement of anisotropic tetrahedral grids under Hessian metrics On a nested refinement of anisotropic tetrahedral grids under Hessian metrics Shangyou Zhang Abstract Anisotropic grids, having drastically different grid sizes in different directions, are efficient and

More information

The Finite Element Method for 2D elliptic PDEs

The Finite Element Method for 2D elliptic PDEs Chapter 9 The Finite Element Method for 2D elliptic PDEs The procedure of the finite element method to solve 2D problems is the same as that for 1D problems, as the flow chart below demonstrates. PDE Integration

More information

Polygonal spline spaces and the numerical solution of the Poisson equation

Polygonal spline spaces and the numerical solution of the Poisson equation Polygonal spline spaces and the numerical solution of the Poisson equation Michael S. Floater, Ming-Jun Lai September 10, 2015 Abstract It is known that generalized barycentric coordinates (GBCs) can be

More information

Scientific Computing: Interpolation

Scientific Computing: Interpolation Scientific Computing: Interpolation Aleksandar Donev Courant Institute, NYU donev@courant.nyu.edu Course MATH-GA.243 or CSCI-GA.22, Fall 25 October 22nd, 25 A. Donev (Courant Institute) Lecture VIII /22/25

More information

The Simplex Algorithm. Chapter 5. Decision Procedures. An Algorithmic Point of View. Revision 1.0

The Simplex Algorithm. Chapter 5. Decision Procedures. An Algorithmic Point of View. Revision 1.0 The Simplex Algorithm Chapter 5 Decision Procedures An Algorithmic Point of View D.Kroening O.Strichman Revision 1.0 Outline 1 Gaussian Elimination 2 Satisfiability with Simplex 3 General Simplex Form

More information

(x,y ) 3 (x,y ) (x,y ) φ 1

(x,y ) 3 (x,y ) (x,y ) φ 1 Chapter 4 Finite Element Approximation 4. Introduction Our goal in this chapter is the development of piecewise-polynomial approximations U of a two- or three-dimensional function u. For this purpose,

More information

Lectures in Discrete Differential Geometry 3 Discrete Surfaces

Lectures in Discrete Differential Geometry 3 Discrete Surfaces Lectures in Discrete Differential Geometry 3 Discrete Surfaces Etienne Vouga March 19, 2014 1 Triangle Meshes We will now study discrete surfaces and build up a parallel theory of curvature that mimics

More information

Commutative filters for LES on unstructured meshes

Commutative filters for LES on unstructured meshes Center for Turbulence Research Annual Research Briefs 1999 389 Commutative filters for LES on unstructured meshes By Alison L. Marsden AND Oleg V. Vasilyev 1 Motivation and objectives Application of large

More information

5. GENERALIZED INVERSE SOLUTIONS

5. GENERALIZED INVERSE SOLUTIONS 5. GENERALIZED INVERSE SOLUTIONS The Geometry of Generalized Inverse Solutions The generalized inverse solution to the control allocation problem involves constructing a matrix which satisfies the equation

More information

On Jeśmanowicz Conjecture Concerning Pythagorean Triples

On Jeśmanowicz Conjecture Concerning Pythagorean Triples Journal of Mathematical Research with Applications Mar., 2015, Vol. 35, No. 2, pp. 143 148 DOI:10.3770/j.issn:2095-2651.2015.02.004 Http://jmre.dlut.edu.cn On Jeśmanowicz Conjecture Concerning Pythagorean

More information

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables

More information

1 Exercise: Heat equation in 2-D with FE

1 Exercise: Heat equation in 2-D with FE 1 Exercise: Heat equation in 2-D with FE Reading Hughes (2000, sec. 2.3-2.6 Dabrowski et al. (2008, sec. 1-3, 4.1.1, 4.1.3, 4.2.1 This FE exercise and most of the following ones are based on the MILAMIN

More information

PITSCO Math Individualized Prescriptive Lessons (IPLs)

PITSCO Math Individualized Prescriptive Lessons (IPLs) Orientation Integers 10-10 Orientation I 20-10 Speaking Math Define common math vocabulary. Explore the four basic operations and their solutions. Form equations and expressions. 20-20 Place Value Define

More information

Mathematical Programming and Research Methods (Part II)

Mathematical Programming and Research Methods (Part II) Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types

More information