Class xlifepp::LagrangeStdSegment#

class LagrangeStdSegment : public xlifepp::LagrangeSegment#

Inheritance diagram for xlifepp::LagrangeStdSegment:

Collaboration diagram for xlifepp::LagrangeStdSegment:

child class of LagrangeHexahedron with standard interpolation

Public Functions

LagrangeStdSegment(const Interpolation *interp_p)#

constructor by interpolation

Lagrange segment Reference Element for Pk interpolation at equidistant nodes.

inline ~LagrangeStdSegment()#

destructor

virtual void computeShapeFunctions()#

compute polynomials representation

compute shape functions as polynomials using general algorithm

virtual void computeShapeValues(std::vector<real_t>::const_iterator it_pt, ShapeValues &shv, const bool withDeriv = true, const bool with2Deriv = false) const#

compute shape functions at given point

shapeFunctions defines Lagrange Reference Element shape functions

Shape functions (Lagrange polynomials at $n_0=1,n_1=0,n_2=1-1/n,n_3=1-2/n,\dots n_n=1/n$) with $w_i(n_j)={\delta }_{ij}(0<=i,j<=n)$, are given by

$\begin{array}{rl}{w}_0& {\displaystyle =\frac{n^{n-1}}{(n-1)!}x\prod _{j=1,n-1}(x-j/n)}\\ & {\displaystyle =\frac{x}{(n-1)!}\prod _{j=1,n-1}(nx-j)}\\ w_1& {\displaystyle =(-1{)}^n{n}^{n-1}{C}_n^n(x-1)\frac{x-1/n}{1}\dots \frac{x-(n-1)/n}{n-1}}\\ & {\displaystyle =(-1{)}^n\frac{n^{n-1}}{(n-1)!}(x-1)\prod _{j=1,n-1}(x-j/n)}\\ & {\displaystyle =(-1{)}^n\frac{x-1}{(n-1)!}\prod _{j=1,n-1}(nx-j)}\\ w_2& {\displaystyle =(-1)n^{n-1}{C}_n^{1}x(x-1)\frac{(x-1/n)}{1}\dots \frac{(x-(j-1)/n)}{(j-1)}\frac{(x-j)/n)}{j}\ast \frac{(x-(j+1)/n)}{(j+1)}\dots \frac{(x-(n-2)/n)}{(n-2)}}\\ & {\displaystyle =(-1)\frac{n^{n-1}}{(n-1)!}\frac{n!}{((n-2)2!)}\prod _{j=0,n\mathrm{\&}j!=n-1}(x-j/n)}\\ & {\displaystyle =(-1)\frac{nx(x-1)}{((n-2)2!)}\prod _{j=1,n-2}(nx-j)}\\ \mathrm{\forall }i=3,n{w}_i& {\displaystyle =(-1{)}^{i-1}{n}^{n-1}{C}_n^{i-1}x(x-1)\frac{(x-1/n)}{1}\dots \frac{(x-(k-1)/n)}{(k-1)}\frac{(x-(k+1)/n)}{(k+1)}\dots \frac{(x-(n-1)/n)}{(n-1)}}\\ & {\displaystyle =(-1{)}^{i-1}\frac{n^{n-1}}{(n-1)!}\frac{n!}{((n-i)i!)}\prod _{j=1,n-1\mathrm{\&}j!=k}(x-j/n)}\\ & {\displaystyle =(-1{)}^{i-1}\frac{n\ast x(x-1)}{((n-i)i!)}\prod _{j=1,n-1\mathrm{\&}j!=k}(n\ast x-j)}\end{array}$

where $k=n+1-i$ and ${\displaystyle C_n^j=\frac{n!}{(n-j)!j!}}$ are Pascal’s triangle binomial coefficients.

Examples:

• n=1

  • $w_0=1\ast 1^{0}x=x$

  • $w_1=-1\ast 1^{(}x-1)=1-x$

• n=2

  • $w_0=1\ast 2^1/1x(x-1/2)=x\ast (2x-1)$

  • $w_1=1\ast 2^1/1(x-1)(x-1/2)=(x-1)\ast (2x-1)$

  • $w_2=-2\ast 2^1/1x(x-1)=-4x(x-1)$

• n=3

  • $w_0=1\ast 3^2/2x(x-1/3)(x-2/3)=1/2x(3x-1)(3x-2)$

  • $w_1=-1\ast 3^2/2(x-1)(x-1/3)(x-2/3)=-1/2(x-1)(3x-1)(3x-2)$

  • $w_2=-3\ast 3^2/2x(x-1)(x-1/3)=-9/2x(x-1)(3x-1)$

  • $w_3=3\ast 3^2/2x(x-1)(x-2/3)=9/2x(x-1)(3x-2)$

  Note: shv has to be sized before

inline virtual const splitvec_t &getO1splitting() const#

returns reference to splitting scheme computed by splitO1

virtual void pointCoordinates()#

point coordinates (!) (equidistant on [0,1])

pointCoordinates defines Lagrange Reference Element point coordinates

Local numbering of Lagrange Pk 1D elements

--1-- 2---1  2---3---1  2---4---3---1  2---5---4---3---1  2---6---5---4---3---1
 k=0   k=1      k=2          k=3              k=4                  k=5