BezierBernsteinMethods
This package implements Bezier-Bernstein methods on simplicial meshes.
References
- Kirby, R.C. Fast simplicial finite element algorithms using Bernstein polynomials. Numer. Math. 117, 631–652 (2011). https://doi.org/10.1007/s00211-010-0327-2
- Mark Ainsworth, Gaelle Andriamaro, and Oleg Davydov. Bernstein-Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures. SIAM J. Sci. Comput. 33, 6 (November 2011), 3087–3109. https://doi.org/10.1137/11082539X
- Kirby, R.C. Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices. SIAM Journal on Scientific Computing 36:2, A846-A868 (2014). https://doi.org/10.1137/130927693
Index
BezierBernsteinMethods.BezierSimplexBezierBernsteinMethods.LinearIndexBezierBernsteinMethods.LinearIndexBezierBernsteinMethods.MultiIndicesBezierBernsteinMethods.SimplexBezierBernsteinMethods.BinomialBezierBernsteinMethods.b2cBezierBernsteinMethods.barycentriccoordsBezierBernsteinMethods.bernsteinBezierBernsteinMethods.bernsteinfunsBezierBernsteinMethods.cartesiancoordsBezierBernsteinMethods.decasteljau!BezierBernsteinMethods.degree_elevation_operatorBezierBernsteinMethods.grevillepointBezierBernsteinMethods.jacobianBezierBernsteinMethods.kformbasisconvBezierBernsteinMethods.linindexBezierBernsteinMethods.mass_matrixBezierBernsteinMethods.multiindexBezierBernsteinMethods.step_decasteljau!BezierBernsteinMethods.subsetsBezierBernsteinMethods.volIgaBase.dimensionIgaBase.dimsplinespace
BezierBernsteinMethods.Binomial — MethodBinomial(Val(n), Val(k)) = binomial(n,k)@generated function for optimized computation of binomials.
BezierBernsteinMethods.b2c — Methodb2c(simplex)Return the mapping (matrix) that maps barycentric coordinates to Cartesian coordinates.
BezierBernsteinMethods.barycentriccoords — Methodbarycentriccoords(S::AbstractSimplex, x)Compute the barycentric coordinates of a point, or matrix of points x.
BezierBernsteinMethods.bernstein — Methodbernstein(p::Degree, control::AbstractVector{T}, λ::BPoint{D,T})Compute Bezier polynomial at barycentric point λ with the degrees of freedom in control.
BezierBernsteinMethods.bernsteinfuns — Methodbernstein(X::MultiIndex{D}, λ::BPoint{D,T})Compute single Bernstein basis function corresponding to multi-index X at barycentric point λ.
BezierBernsteinMethods.cartesiancoords — Methodcartesiancoords(S::AbstractSimplex, λ)Compute the Cartesian coordinates of a barycentric point or vector of barycentric points λ.
BezierBernsteinMethods.decasteljau! — Methoddecasteljau!(control::Vector{T}, λ::BPoint{D,T}, MultiIndices)Classic DeCasteljau algorithm (can compute derivatives too!) on a (D-1)-dimensional simplex.
BezierBernsteinMethods.degree_elevation_operator — Functiondegree_elevation_operator(p::Degree, d::Dimension)Return a matrix that maps degrees of freedom of a Bezier polynomial of degree p to the degrees of freedom corresponding to its order elevated Bezier polynomial.
BezierBernsteinMethods.grevillepoint — Methodgrevillepoint(X::MultiIndex)Compute Greville point associated with multi-index label X.
BezierBernsteinMethods.jacobian — Methodjacobian(simplex)Return the Jacobian mapping of a simplex.
BezierBernsteinMethods.kformbasisconv — Methodkformbasisconv(n::Dimension, k::Form, r::Degree)Compute the linear mapping from Bernstein polynomials to the space of k-forms Pᵣ⁻Λᵏ(T). This linear mapping is encoded in Pattern{T}. We refer to the paper. For more information, check out the periodic table of finite elements
BezierBernsteinMethods.linindex — Methodlinindex(i, j, k...)Return the linear index of a multi-index label.
BezierBernsteinMethods.mass_matrix — Methodmass_matrix(S::AbstractSimplex{D}, p::Degree, q::Degree, T::Type{<:Real} = Type{Float64})Compute Bernstein mass matrix for degree (p,q) and dimension d.
BezierBernsteinMethods.multiindex — MethodMultiIndex(::LinearIndex{P<:Degree, D<:Dimension, I<:Integer})Return the multi-index associated with linear index I and simplicial domain of degree P and dimension D.
BezierBernsteinMethods.step_decasteljau! — Methodstep_decasteljau!(control::Vector{T}, λ::BPoint{D,T}, MultiIndices)Computes a single step of the recursive DeCasteljau algorithm on a (D-1)-dimensional simplex.
BezierBernsteinMethods.subsets — Methodsubsets(v::NTuple{n,Int}, k::Int)Generate the binomial(n,k) unique subsets of length k from set v and its complement.
Examples:
julia> α, β = subsets((1,2,3), 1);
julia> α
3-element Array{Tuple{Int64},1}:
(1,)
(2,)
(3,)
julia> β
3-element Array{Tuple{Int64,Int64},1}:
(2, 3)
(1, 3)
(1, 2)BezierBernsteinMethods.vol — Methodvol(simplex)Compute the volume of an affine simplex.
IgaBase.dimension — Methoddimension(simplex)Return the dimension of a simplex.
IgaBase.dimsplinespace — Methoddimsplinespace(p::Degree, d::Dimension)Compute the dimension of the Bezier spline space on a simplex of degree p and numer of vertices d.
BezierBernsteinMethods.BezierSimplex — Type BezierSimplex{D,T<:Real} <: AbstractSimplex{D}Simplex of dimension D with Vertices in real number type T.
BezierBernsteinMethods.LinearIndex — TypeLinearIndex(P::Degree, D::Dimension, I::Integer)A linear index I of a simplicial discretization with Degree P and dimension D.
BezierBernsteinMethods.LinearIndex — MethodLinearIndex(::MultiIndex{X})Compute the LinearIndex associated with a MultiIndex.
BezierBernsteinMethods.MultiIndices — TypeMultiIndices(P::Degree, D::Dimension)
MultiIndices(D::Dimension)
MultiIndices(s::Simplex)
MultiIndices(s::BezierSimplex)Returns an iterator corresponding to a Bezier simplex of polynomial degree P and dimension D.
BezierBernsteinMethods.Simplex — Type Simplex{D,T<:Real} <: AbstractSimplex{D}Simplex of dimension D with Vertices in real number type T.