Internals

ϵ (\epsilon)

an alias for Base.eps(𝕣).

∞ (\infty)

an alias for Base.inf.

An "identity vector"

id = IdVec i = 9834987 id[i] == i # true for any i

See also Julia's identity function.

A simple offset vector that only implements "setindex!" and "getindex". Based on Memory, so cannot be extended.

Muscade.addin!(asm,global,block,ibr,ibc,factor=1.)

Add a sparse block into a large out sparse matrix, at block-row and -column ibr and ibc. Use prepare to allocate memory for global and build the assembler asm.

addin!(asm,outvec,blockvec,ibr)

Add a full block vector into a large outvec full vector. at block-row ibr. Use prepare to create asm.

See also: prepare

λ,v,ncv = Muscade.geneig{ALGO}(A,B,neig=5)

Solves (A-λ*B)*v=0, finding the neig lowest eigenvalues λ (in absolute value) and the corresponding eigenvectors v (a Vector{Vector})

Input

ALGO can be

• :SDP if A is symmetric definite positive and B is symmetric. Will return real λ and v.
• :Hermitian if A is symmetric indefinite and B is symmetric. Will return real λ and v.
• :Complex otherwise, will return complex λ and v.

Optional keyword arguments:

• maxiter = 300
• verbosity = 0 ∈ {0,1,2,3}
• krylovdim = 2neig+6

Uses KrylovKit.jl. Freely based on VibrationGEPHelpers.jl and input from PetrKryslUCSD and stevengj. See GIThub-blame for bug-credits.

to_order{P,N}(V)

Decrease (lossy) or increase (pad partials with zeros) the order of differentiation of V. V is a nested structure of NamedTuple, Tuple, SArray, and the components of V must be of type ∂ℝ or 𝕣.

IMPORTANT:

1. assumes all the nested differentiations are with respect to the same variable.
2. to_order{P,N}(V::𝕣) = V

X = variate_{P,N}(x)

where typeof(x)<:SVector{N}, create a SVector of automatic differentiation objects of precedence P.

X = variate_{P}(x)

where typeof(x)<:Real, create an object of precedence P.

!!! Warning Improper use may cause perturbation confusion. Assumes, without testing, that precedence(x)=P-1

See also: precedence, δ_, value, ∂, VALUE, value_∂

X = δ_{P,N,R}()

create a SVector of automatic differentiation objects of precedence P and value zero.

X = δ_{P}()

Create automatic differentiation object of precedence P and value zero.

!!! Warning Improper use may cause perturbation confusion

See also: precedence, Muscade.variate_, value, ∂, VALUE, value_∂

ℕ (\bbN)

an alias for UInt64. For use in dispatching. ℕ1... ℕ4 are AbstractArrays of dimensions 1 to 4.

𝕔 (\bbc)

an alias for Complex{Float64}. For use in struct definitions. 𝕔1... 𝕔4 are Arrays of dimensions 1 to 4. 𝕔11 is a Vector of Vector.

𝕟 (\bbn)

an alias for UInt64. For use in struct definitions. 𝕟1... 𝕟4 are Arrays of dimensions 1 to 4.

s2 = copy(s1::State)

Copy a state so that s1 and s2 do not share memory.

(Actualy the States will stil share the same Model, but no operations are provided to mutate it).

s2 = copy(s1;[time=t])

create a copied State with time t.

See also: initialize!, setdof!

colnormalize(a)

Euclidian-normalize the columns of an SMatrix

columnmatrix(v)

Reshape a vector into a matrix of size (length(v),1)

et = eletyp(model)

Return a vector of the concrete types of elements in the model.

str = indent(str,n)

Add n spaces at the begining of every line in str.

Muscade.isapprox_dbg(a,b;kwargs...)

Compare two ∂ℝ of the same type (or two conformant AbstractArrays of same eltype). The comparison applies to the value and the partial derivatives.

This differs from

isapprox(a,b;kwargs...)

which only compares values, in keeping with the principle that ∂ℝ must behave as ℝ.

bigmat,bigmatasm,bigvecasm,bigvecdis = Muscade.prepare(pattern)

Prepare for the assembly of sparse blocks into a large sparse matrix. bigmat is allocated, with the correct sparsity structure, but its nzval undef'ed. Where some blocks share the same sparsity structure, blocks in pattern can have === elements.

pattern is a SparseMatrixCSC{<:SparseMatrixCSC}, where empty blocks are structuraly zero

See also: addin!

rowmatrix(v)

Reshape a vector into a matrix of size (1,length(v))

Muscade.sinc1(x)

sinc1(x) = sin(x)/x - but sinc1(0.) = 1.. The function can be differentiated to the fourth order.

This differs from Julia's sinc(x) = sin(π*x)/(π*x).

sparser!(S::SparseMatrixCSC,keep::Function)

Eliminate terms that do not satisfy a criteria from the storage of a sparse matrix. S will be mutated, and the size of its internal storage modified. keep is a Function which to an index into S.nzval associate true if storage is to be kept for this term and false otherwise. Alternatively, keep can be a Vector{Bool}

Examples

sparser!([T],S,i->abs(S.nzval[i])>tol)
sparser!([T],S,keep::Vector{Boolean})

If T is provided (initialised as T = copy(S)), then result is set in T, S is unchanged, other wise S is mutated in place. The input keep::Vector{Boolean} must be of length nnz(S).

sparser!([S1,S2,...],rtol=1e-9)

Operates in place, reducing the sparses S1, S2 etc... to a common sparsity pattern.

sA = symmetric!(A)

Compute the symmetric part of a square Matrix. Mutates the input argument.

iV   = variate_indices(V)

For use in conjunction with

TV   = variate(V)

iV has the same structure as V and TV but contains integers: the indices into the partials of TV

See also: revariate

zero!(a)

Set to zero all elements of an arrays. If a is sparse, the vector nzval of values is set to zero and the sparsity structure is unchanged.

𝕫log2(i::𝕫)

Compute the integer log2 of an integer, fast. Fails if i is not a power of two.

trace(v::SMatrix{3,3})

Computes the trace of a matrix.

vec3(v, ind)

Create a 3-element static vector from selected elements of vector v.

Arguments

• v: The input vector from which elements are selected.
• ind: A collection of 3 indices specifying which elements of v to include.

Returns

• An SVector{3} containing the elements v[ind[1]], v[ind[2]], v[ind[3]].

Drawing a Bar3D.

draw!(axis,state)

Optional arguments (and their default values) are

• line_color = :black color of the line
• Udof (true iff element has Udofs) wether to draw U-forces.
• Uscale = 1. How many meter is a Newton per meter?

Drawing a EulerBeam3D.

draw!(axis,state)

draw!(axis,state;EulerBeam3D=(;style=:shape))

α      = 2π*(0:19)/20
circle = 0.1*[cos.(α) sin.(α)]'
draw!(axis,state;EulerBeam3D=(;style=:solid,section = circle))

style=:shape shows the deformed neutral axis of the element. It has optional arguments frame=true (draws the element's corotated frame of reference) and nseg=10 (number of points to show the deflected shape of each element).

style=:solid shows the deformed shape of the element. It requires the input section=... to be given a matrix of size (2,nsec) describing nsec points around the cross section of the element (no need to close the circumference by repeating the first point at the end). It has optional arguments nseg=10 as above, marking=true to draw a longitudinal marking and solid_color=:yellow.

Other optional arguments (and their default values) are

• Udof (true iff element has Udofs) wether to draw U-forces.
• draw_frame = false wether to draw the local reference frame of each element
• draw_marking = true wether to draw "longitudinal marking" along the element. Will only draw if style=:solid.
• nseg = 1 number of segments to display the shape of a deformed element
• solid_color = :yellow color of the surface if style=:solid
• line_color = :black color of the line if style=:sshape
• Uscale = 1. How many meter is a Newton per meter?