Coreform IGA Reference Manual

$\newcommand{\bcdot}{\boldsymbol{\cdot}} \newcommand{\wildcard}{*} \newcommand{\Detailed}[3]{{#1}_{#2}^{#3}} \newcommand{\Of}[2]{{#1}(#2)} \newcommand{\From}[2]{{#1}[#2]} \newcommand{\ParentOfChild}[2]{#2^{#1}} \newcommand{\ChildOfParent}[2]{#1^{#2}} \newcommand{\RestrictedBy}[2]{{#1}\vert_{#2}} \newcommand{\Parent}{\operatorname{parent}} \newcommand{\ParentOf}[1]{\Of{\Parent}{#1}} \newcommand{\Children}{\operatorname{children}} \newcommand{\ChildrenOf}[1]{\Of{\Children}{#1}} \newcommand{\Neighborhood}{\operatorname{nbhd}} \newcommand{\NeighborhoodOf}[1]{\Of{\Neighborhood}{#1}} \newcommand{\Skeleton}{\operatorname{skel}} \newcommand{\SkeletonOf}[1]{\Of{\Skeleton}{#1}} \newcommand{\DomainOf}[1]{\mathop{\textrm{domain}}(#1)} \newcommand{\Interior}{\mathop{\textrm{interior}}} \newcommand{\Distance}{\operatorname{dist}} \newcommand{\SuchThat}{\, : \,} \newcommand{\SuchThatText}{\text{s.t.}} \newcommand{\DefEq}{\overset{\operatorname{def}}{=}} \newcommand{\True}{\textsf{True}} \newcommand{\False}{\textsf{False}} \newcommand{\OR}{\operatorname{OR}} \newcommand{\AND}{\operatorname{AND}} \newcommand{\XOR}{\operatorname{XOR}} \newcommand{\ArrayRoster}[1]{\left[#1\right]} \newcommand{\ArrayRosterI}[2]{[#1]_{#2}} \newcommand{\ArrayEmpty}{\ArrayRoster{\hspace{1mm}}} \newcommand{\Tuple}[1]{\mathsf{#1}} \newcommand{\TupleRoster}[1]{\left(#1\right)} \newcommand{\PointPair}[2]{\TupleRoster{#1, #2}} \newcommand{\Set}[1]{\mathsf{#1}} \newcommand{\SetRoster}[1]{\left\{ #1 \right\}} \newcommand{\SetRosterIndexed}[3]{\SetRoster{#1}_{#2}^{#3}} \newcommand{\SetRosterPredicate}[2]{\SetRoster{#1 \SuchThat #2}} \newcommand{\SetEmpty}{\varnothing} \newcommand{\SetPower}[1]{\Of{\boldsymbol{\Set{P}}}{#1}} \newcommand{\SetNeighborhoodOf}[1]{\Of{\boldsymbol{\Set{B}}}{#1}} \newcommand{\Domain}{\Omega} \newcommand{\DomainBdry}{\Gamma} \newcommand{\IntervalClosed}[2]{\left[#1, #2\right]} \newcommand{\IntervalOpen}[2]{\left(#1, #2\right)} \newcommand{\IntervalClosedOpen}[2]{\left[#1, #2\right)} \newcommand{\IntervalOpenClosed}[2]{\left(#1, #2\right]} \newcommand{\Enum}{\operatorname{enum}} \newcommand{\EnumOf}[2]{\Of{\Enum}{#1,#2}} \newcommand{\ContiguousIndexMap}[1]{i_{#1}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Naturals}{\mathbb{N}} \newcommand{\Integers}{\mathbb{Z}} \newcommand{\Field}{\mathbb{F}} \newcommand{\Size}[1]{\left\lvert #1 \right\rvert} \newcommand{\Union}[3]{\bigcup\limits_{#1}^{#2} #3} \newcommand{\UnionInline}[3]{\bigcup_{#1}^{#2} #3} \newcommand{\Intersect}[3]{\bigcap\limits_{#1}^{#2} #3} \newcommand{\ElementWiseGreaterOrEqual}{\succeq} \newcommand{\ElemntWiseGreater}{\succ} \newcommand{\ElementWiseLessOrEqual}{\preceq} \newcommand{\ElementWiseLess}{\prec} \newcommand{\EquivalenceClass}{\varpi} \newcommand{\EquivalenceClassDetailed}[2]{\Detailed{\EquivalenceClass}{#1}{#2}} \newcommand{\EquivalenceClassParallel}[1]{\EquivalenceClassDetailed{#1}{\parallel}} \newcommand{\EquivalenceClassPerp}[1]{\EquivalenceClassDetailed{#1}{\perp}} \newcommand{\EquivalenceClassStar}[1]{\EquivalenceClassDetailed{#1}{*}} \newcommand{\SpaceLinear}[1]{\mathnormal{#1}} \newcommand{\SpaceDual}[1]{\SpaceLinear{#1}^*} \newcommand{\BoundaryOp}{\mathop{\partial}} \newcommand{\AbsoluteValue}[1]{\Size{#1}} \newcommand{\Norm}[2]{\left\lVert #2 \right\rVert_{#1}} \newcommand{\NormAbstract}[1]{\Norm{}{#1}} \newcommand{\NormSemi}[1]{\Size{#1}} \newcommand{\SpaceLinearNormed}[1]{\left(\SpaceLinear{#1}, \NormAbstract{\bcdot} \right)} \newcommand{\SpaceBanach}[1]{\mathfrak{#1}} % normed and complete \newcommand{\SymmetricBilinear}[2]{\left(#1, #2\right)} \newcommand{\InnerProduct}[2]{\left\langle#1, #2\right\rangle} \newcommand{\DotProduct}[2]{\left(#1 \bcdot #2\right)} \newcommand{\SpaceInnerProduct}[1]{\left(\SpaceLinear{#1}, \InnerProduct{\bcdot}{\bcdot}\right)} \newcommand{\SpaceHilbert}[1]{\mathcal{#1}} \newcommand{\SpaceLTwo}{\SpaceHilbert{L}_2} \newcommand{\SpaceLTwoDef}[2]{\SpaceLTwo\left(#1; #2\right)} \newcommand{\SpaceLInfty}{\SpaceHilbert{L}_\infty} \newcommand{\SpaceLTwoWeighted}{\SpaceLTwo^w} \newcommand{\SpaceSobolevK}[1]{\SpaceHilbert{H}^{#1}} \newcommand{\SpaceSobolevKDef}[3]{\SpaceSobolevK{#1}\left(#2 ; #3\right)} \newcommand{\SpaceFunction}{\SpaceHilbert{N}} \newcommand{\SpaceFunctionTest}{\SpaceHilbert{U}} \newcommand{\SpaceFunctionTrial}{\SpaceHilbert{V}} \newcommand{\Cont}{\operatorname{cont}} \newcommand{\SpaceCont}{\SpaceHilbert{C}} \newcommand{\SpaceContDef}[2]{\SpaceCont\left(#1 ; #2\right)} \newcommand{\SpaceContK}[1]{\SpaceCont^{#1}} \newcommand{\SpaceContKDef}[3]{\SpaceContK{#1}\left(#2; #3\right)} \newcommand{\SpaceContKBounded}[2]{\SpaceContK{#1}_{#2}} \newcommand{\SpaceContKBoundedDef}[3]{\SpaceCont_{b}^{#1}\left(#2 ; #3\right)} \newcommand{\SpaceContInf}{\SpaceContK{\infty}} \newcommand{\SpaceContInfDef}[2]{\SpaceContKDef{\infty}{#1}{#2}} \newcommand{\SpaceContZero}{\SpaceContK{0}} \newcommand{\SpaceContDisc}{\SpaceContK{-1}} \newcommand{\SpaceLinearOperator}[1]{\mathscr{#1}} \newcommand{\SpaceLinearOperatorDef}[2]{\SpaceLinearOperator{L}\left(#1 ; #2\right)} \newcommand{\SpaceLinearOperatorBoundedDef}[2]{\SpaceLinearOperator{B}\left(#1 ; #2\right)} \newcommand{\SpaceDim}{n} \newcommand{\SpaceLinearN}[1]{\SpaceLinear{#1}^h} \newcommand{\SpaceHilbertN}[1]{\SpaceHilbert{#1}^h} \newcommand{\SpaceFunctionN}{\SpaceFunction^h} \newcommand{\SpaceFunctionTestN}{\SpaceFunctionTest^h} \newcommand{\SpaceFunctionTrialN}{\SpaceFunctionTrial^h} \newcommand{\SpaceFunctionTestDim}{m} \newcommand{\SpaceFunctionTrialDim}{n} \newcommand{\SpaceMatrixRealMN}[2]{\Reals^{#1\times#2}} \newcommand{\SpaceVecRealN}[1]{\Reals^{#1}} \newcommand{\SpacePoly}[1]{\SpaceHilbert{P}^{#1}} \newcommand{\SpacePolyDef}[3]{\SpacePoly{#1}\left(#2 ; #3\right)} \newcommand{\SpaceSpline}{\SpaceHilbert{S}} \newcommand{\LinearOperator}[1]{\SpaceLinearOperator{#1}} \newcommand{\LinearOperatorDef}[3]{\FuncDef{\LinearOperator{#1}}{#2}{#3}} \newcommand{\Functional}[1]{\LinearOperator{#1}} \newcommand{\Projection}{\Pi} \newcommand{\ProjectionDetailed}[2]{\Projection_{#1}^{#2}} \newcommand{\ProjectionParallel}[1]{\ProjectionDetailed{#1}{\parallel}} \newcommand{\ProjectionParallelOf}[2]{\Of{\ProjectionParallel{#1}}{#2}} \newcommand{\ProjectionPerp}[1]{\ProjectionDetailed{#1}{\perp}} \newcommand{\ProjectionPerpOf}[2]{\Of{\ProjectionPerp{#1}}{#2}} \newcommand{\ProjectionStar}[1]{\ProjectionDetailed{#1}{*}} \newcommand{\ProjectionStarOf}[2]{\Of{\ProjectionStar{#1}}{#2}} \newcommand{\MacBracket}[1]{\langle#1\rangle} \newcommand{\BraKet}[1]{\langle#1\rangle} \newcommand{\HeavisideFunc}[1]{\mathit{H}\left(#1\right)} \newcommand{\DiracDelta}[2]{\delta_{D}\left(#1 - #2\right)} \newcommand{\Func}[1][f]{#1} \newcommand{\FuncAlt}[1][g]{#1} \newcommand{\FuncDef}[3]{#1 \SuchThat #2 \rightarrow #3} \newcommand{\FuncApprox}[1]{#1^h} \newcommand{\erf}[1]{\mathrm{erf}\left(#1\right)} \newcommand{\erfc}[1]{\mathrm{erfc}\left(#1\right)} \newcommand{\Mat}[1]{\boldsymbol{#1}} \newcommand{\MatIJ}[3]{\left[#1\right]_{#2#3}} \newcommand{\MatIdRow}{m} \newcommand{\MatIdCol}{n} \newcommand{\ColumnSelector}{\operatorname{\mathrm{col}}} \newcommand{\RowSelector}{\operatorname{\mathrm{row}}} \newcommand{\ColOf}[1]{\ColumnSelector_{#1}} \newcommand{\Cols}[1]{\mathop{\mathrm{cols}}{#1}} \newcommand{\RowOf}[1]{\RowSelector_{#1}} \newcommand{\Rows}[1]{\mathop{\mathrm{rows}}{#1}} \newcommand{\DiagonalMatrix}[1]{\Of{\operatorname*{diag}}{#1}} \newcommand{\Rotation}{\Mat{R}} \newcommand{\Identity}{\Mat{I}} \newcommand{\KroneckerDeltaComp}{\delta} \newcommand{\KroneckerDelta}{\Mat{\delta}} \newcommand{\LeviCivita}{\Mat{\epsilon}} \newcommand{\ZeroMat}{\Mat{0}} \newcommand{\IdentityTenSym}{\mathbb{I}} \newcommand{\GramianMat}{\Mat{G}} \newcommand{\Gramian}[2]{\ParentOfChild{#1}{\Of{\Mat{G}}{#2}}} \newcommand{\TraceMappingMat}{\Mat{M}} \newcommand{\ChangeOfBasisMat}[1]{\Mat{T}^{#1}} \newcommand{\OrthogonalMat}{\Mat{Q}} \renewcommand{\Vec}[1]{\boldsymbol{#1}} \newcommand{\VecChildOf}[2]{\ParentOfChild{#2}{#1}} \newcommand{\VecI}[2]{\{#1\}_{#2}} \newcommand{\ZeroVec}{\Vec{0}} \newcommand{\OneVec}{\Vec{1}} \newcommand{\UnitNormalComp}{n} \newcommand{\UnitNormal}{\Vec{\UnitNormalComp}} \newcommand{\NonZeroIndices}{\operatorname{\mathsf{nzi}}} \newcommand{\NonZeroIndicesOf}[1]{\Of{\NonZeroIndices}{#1}} \newcommand{\Reindex}[2]{\mathop{\mathrm{reindex}_{#1}{#2}}} \newcommand{\FuncObjective}[1]{\Of{\Func}{#1}} \newcommand{\VarDesign}{\Vec{\phi}} \newcommand{\VarState}[1]{\Of{\Vec{u}}{#1}} \newcommand{\FuncLagrangian}[1]{\Of{\mathscr{L}}{#1}} \newcommand{\FuncAugmentedLagrangian}[1]{\Of{\mathscr{L}_A}{#1}} \newcommand{\FuncEqualityConstraints}[1]{\Of{\Func[\Vec{g}]}{#1}} \newcommand{\PenaltyParameter}{\rho} \newcommand{\Prod}[2]{\prod\limits_{#1}^{#2}} \newcommand{\ProdInline}[2]{\prod_{#1}^{#2}} \newcommand{\TensorProduct}{\otimes} \newcommand{\Sum}[2]{\sum\limits_{#1}^{#2}} \newcommand{\SumInline}[2]{\sum_{#1}^{#2}} \newcommand{\DirectAdd}[2]{{#1}\oplus{#2}} \newcommand{\DirectSum}[2]{\bigoplus_{#1}{#2}} \newcommand{\Rank}[1]{\operatorname{rank}(#1)} \newcommand{\Nullity}[1]{\operatorname{null}(#1)} \newcommand{\Image}{\operatorname{image}} \newcommand{\Range}{\operatorname{range}} \newcommand{\Inverse}[1]{\left(#1\right)^{-1}} \newcommand{\inverse}[1]{\bar{#1}} \newcommand{\Transpose}[1]{\left(#1\right)^{T}} \newcommand{\TransposeSimple}[1]{#1^{T}} \newcommand{\InvTrans}[1]{\left(#1\right)^{-T}} \newcommand{\Complement}[1]{\bar{#1}} \newcommand{\Adjoint}[1]{\left(#1\right)^*} \newcommand{\OrthogonalComplement}[1]{\left(#1\right)^\perp} \newcommand{\Det}[1]{\Of{\operatorname{det}}{#1}} \newcommand{\DCT}{\operatorname{DCT}} \newcommand{\IDCT}{\operatorname{IDCT}} \newcommand{\RealPart}[1]{\Of{\mathfrak{R}}{#1}} \newcommand{\Trace}[1]{\Of{\operatorname{tr}}{#1}} \newcommand{\Dev}[1]{\Of{\operatorname{dev}}{#1}} \newcommand{\DevDef}[1]{\Dev{#1} \DefEq #1 - \frac{1}{3}\Trace{#1}\Identity} \newcommand{\Flattening}[1]{\Phi_{#1}} \newcommand{\Scatter}[1]{\Mat{S}^{#1}} \newcommand{\Gather}[1]{\Mat{G}^{#1}} \newcommand{\Coeff}[2]{{#1}_{#2}} \newcommand{\BasisVec}[2]{{#1}_{#2}} \newcommand{\BasisVecChildOf}[3]{\Detailed{#1}{#2}{#3}} \newcommand{\OrthogonalBasisVec}[2]{\Detailed{#1}{#2}{\perp}} \newcommand{\OrthonormalBasisVec}[2]{\Detailed{\bar{#1}}{#2}{\perp}} \newcommand{\BasisFuncId}{i} \newcommand{\BasisFuncIdAlt}{j} \newcommand{\BasisFuncIdAltAlt}{k} \newcommand{\BasisFunc}[3]{\Detailed{#1}{#2}{#3}} \newcommand{\BasisFuncSubscripted}[1]{{N}_{#1}} \newcommand{\BasisFuncTestId}{A} \newcommand{\BasisFuncTest}{M} \newcommand{\BasisFuncTestSubscripted}[1]{\BasisFuncTest_{#1}} \newcommand{\BasisFuncTestUni}[2]{\BasisFuncTestSubscripted{\BasisFuncTestId_{#1}}^{(\delta_{#1#2})}(s_{q_{#1}})} \newcommand{\BasisFuncTrial}{N} \newcommand{\BasisFuncTrialId}{B} \newcommand{\BasisFuncTrialSubscripted}[1]{\BasisFuncTrial_{#1}} \newcommand{\BasisFuncTrialUni}[2]{\BasisFuncTrialSubscripted{\BasisFuncTrialId_{#1}}^{(\delta_{#1#2})}(s_{q_{#1}})} \newcommand{\BasisFuncInSubspace}{\bar{N}} \newcommand{\BasisFuncTestInSubspace}{\bar{\BasisFuncTest}} \newcommand{\BasisFuncTrialInSubspace}{\bar{\BasisFuncTrial}} \newcommand{\BasisFuncInSubspaceSubscripted}[1]{\BasisFuncInSubspace_{#1}} \newcommand{\BasisFuncTestInSubspaceSubscripted}[1]{\BasisFuncTestInSubspace_{#1}} \newcommand{\BasisFuncTrialInSubspaceSubscripted}[1]{\BasisFuncTrialInSubspace_{#1}} \newcommand{\Dual}[1]{{\bar{#1}}} \newcommand{\SymBasisFunc}{{S}} \newcommand{\SymBasisFuncDetailed}[2]{{S}_{{#1}{#2}}} \newcommand{\SymBasisFuncId}[1]{{#1}} \newcommand{\SymDualBasisFuncDetailed}[2]{\Dual{S}_{{#1}{#2}}} \newcommand{\PolynomialDegree}{p} \newcommand{\PolynomialDegreeAlt}{q} \newcommand{\PolynomialBasisFuncNoArg}[1]{\Detailed{\Func[P]}{#1}{}} \newcommand{\PolynomialBasisFunc}[2]{\Of{\PolynomialBasisFuncNoArg{#1}}{#2}} \newcommand{\LagrangeBasisFuncNoArg}[2]{\Detailed{\Func[L]}{#1}{#2}} \newcommand{\LagrangeBasisFunc}[3]{\Of{\Detailed{\Func[L]}{#1}{#2}}{#3}} \newcommand{\BernsteinBasisFuncNoArg}[1]{\Detailed{\Func[B]}{#1}{}} \newcommand{\BernsteinBasisFuncDegNoArg}[2]{\Detailed{\Func[B]}{#1}{#2}} \newcommand{\BernsteinBasisFunc}[2]{\Of{\BernsteinBasisFuncNoArg{#1}}{#2}} \newcommand{\BernsteinBasisFuncDeg}[3]{\Of{\BernsteinBasisFuncDegNoArg{#1}{#2}}{#3}} \newcommand{\ChebyshevBasisFuncNoArg}[1]{\Detailed{\Func[T]}{#1}{}} \newcommand{\ChebyshevBasisFunc}[2]{\Of{\ChebyshevBasisFuncNoArg{#1}}{#2}} \newcommand{\ChebyshevPointSymbol}{\chi} \newcommand{\ChebyshevPoint}[1]{\ChebyshevPointSymbol_{#1}} \newcommand{\ChebyshevPointFirst}[1]{\ChebyshevPointSymbol^{(1)}_{#1}} \newcommand{\ChebyshevPointSecond}[1]{\ChebyshevPointSymbol^{(2)}_{#1}} \newcommand{\BasisVecI}{\Vec{i}} \newcommand{\BasisVecJ}{\Vec{j}} \newcommand{\BasisVecK}{\Vec{k}} \newcommand{\VecExpandIJK}[1]{\DotProduct{#1}{\BasisVecI} \BasisVecI + \DotProduct{#1}{\BasisVecJ} \BasisVecJ + \DotProduct{#1}{\BasisVecK} \BasisVecK} \newcommand{\Span}{\operatorname{span}} \newcommand{\Dim}[1]{\operatorname{dim}\left({#1}\right)} \newcommand{\Support}[1]{\operatorname{supp}(#1)} \newcommand{\SupportPos}[1]{\operatorname{supp}_{+}(#1)} \newcommand{\SumBasis}[4]{\Sum{#1}{#2}{#3}_{#1}{#4}_{#1}} \newcommand{\SumInlineBasis}[4]{\SumInline{#1}{#2}{#3}_{#1}{#4}_{#1}} \newcommand{\BasisTransMatrix}[2]{\Mat{T}_{{#1}\rightarrow{#2}}} \newcommand{\BernToChebMat}{\BasisTransMatrix{B}{C}} \newcommand{\ChebToBernMat}{\BasisTransMatrix{C}{B}} \newcommand{\ChebToLagMat}{\BasisTransMatrix{C}{L}} \newcommand{\LagToChebMat}{\BasisTransMatrix{L}{C}} \newcommand{\BernToLagMat}{\BasisTransMatrix{B}{L}} \newcommand{\LagToBernMat}{\BasisTransMatrix{L}{B}} \newcommand{\ChebyshevZeros}[1]{z_{#1}} \newcommand{\ChebyshevExtremizers}[1]{t_{#1}} \newcommand{\EvaluateTwoLimits}[3]{\left.{#1}\right\rvert_{#2}^{#3}} \newcommand{\EvaluateOneLimit}[2]{\left.{#1}\right\rvert_{#2}} \newcommand{\Limit}[2]{\lim\limits_{#1\rightarrow #2}} \newcommand{\LimitInline}[2]{\lim_{#1\rightarrow #2}} \newcommand{\DNE}{\operatorname{DNE}} \newcommand{\TotalDeriv}{D} \newcommand{\LagrangeDeriv}[1]{{#1}'} \newcommand{\KthLagrangeDeriv}[2]{{#1}^{(#2)}} \newcommand{\LeibnizSubDeriv}[2]{{#1}_{,#2}} \newcommand{\LeibnizFracDeriv}[2]{\frac{d}{d#2} #1} \newcommand{\KthLeibnizFracDeriv}[3]{\frac{d^{#2}}{d#3^{#2}}\Func[#1](#3)} \newcommand{\KthLeibnizFracSpecificFunction}[3]{\frac{d^{#2}}{d#3^{#2}}\Func[#1]} \newcommand{\Differential}[1][x]{\, \operatorname{d}{#1}} \newcommand{\FrechetDeriv}[2]{D{#1}\left(#2\right)} \newcommand{\GateauxDirectionBold}[1]{\Vec{\delta}#1} \newcommand{\GateauxDirection}[1]{\delta #1} \newcommand{\GateauxDerivBold}[2]{D#1(#2; \GateauxDirectionBold{#2})} \newcommand{\GateauxDeriv}[2]{D#1(#2; \GateauxDirection{#2})} \newcommand{\GateauxDerivWithDirection}[3]{D#1\left(#2 \ ;\ #3\right)} \newcommand{\GateauxDerivPartial}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\KthGateauxDerivPartial}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} \newcommand{\Grad}[2]{\boldsymbol{\nabla}^{#1} #2} \newcommand{\Div}[2]{\boldsymbol{\nabla}^{#1} \bcdot #2} \newcommand{\Laplace}[2]{\boldsymbol{\Delta}^{#1} #2} \newcommand{\TaylorSeries}[2]{\mathcal{T}\left[\Func[#1]\right](#2)} \newcommand{\TruncatedTaylorSeries}[3]{\mathcal{T}_{#1}\left[\Func[#2]\right]\left(#3\right)} \newcommand{\JacComp}{J} \newcommand{\JacMat}{\Mat{J}} \newcommand{\JacAdjComp}{\JacComp^{\oslash}} \newcommand{\JacAdjMat}{\JacMat^{\oslash}} \newcommand{\JacInvComp}{\JacComp^{-1}} \newcommand{\JacInvMat}{\JacMat^{-1}} \newcommand{\JacDet}{J} \newcommand{\Int}[2]{\int\limits_{#1}^{#2}} \newcommand{\IntDomain}[1]{\int\limits_{#1}} \newcommand{\IntInline}[2]{\int_{#1}^{#2}} \newcommand{\IntDomainInline}[1]{\int_{#1}} \newcommand{\BigO}[1]{\mathcal{O}\left(#1\right)} \newcommand{\Flops}{\operatorname{flops}} \newcommand{\Interpolant}[2]{\mathcal{#1}^{#2}} \newcommand{\ErrorApproxComp}{e} \newcommand{\ErrorApproxTen}{\Vec{\ErrorApproxComp}} \newcommand{\ErrorNewton}{\epsilon} \newcommand{\Tolerance}{\operatorname{tol}} \newcommand{\Truncate}{\operatorname{trunc}}$$\newcommand{\SymbolDim}{d} \newcommand{\SymbolParentModifier}[1]{\bar{#1}} \newcommand{\SymbolParamModifier}[1]{\hat{#1}} \newcommand{\SymbolSubmeshModifier}{} \newcommand{\SymbolLocalBasis}{B} \newcommand{\SymbolGlobalBasis}{N} \newcommand{\SymbolCoeff}{c} \newcommand{\SymbolUspline}{U} \newcommand{\SymbolUsplineLower}{u} \newcommand{\ParentDomain}{\SymbolParentModifier{\Domain}} \newcommand{\ParentDomainDetailed}[2]{\Detailed{\bar{\Domain}}{#1}{#2}} \newcommand{\ParentDomainBdry}{\SymbolParentModifier{\DomainBdry}} \newcommand{\ParentDomainBdryDetailed}[2]{\Detailed{\bar{\DomainBdry}}{#1}{#2}} \newcommand{\ParentDomainChart}{\SymbolParentModifier{\Vec{\phi}}} \newcommand{\ParentDim}{\SymbolParentModifier{\SymbolDim}} \newcommand{\ParentCoord}{\xi} \newcommand{\ParentS}{\ParentCoord_0} \newcommand{\ParentT}{\ParentCoord_1} \newcommand{\ParentU}{\ParentCoord_2} \newcommand{\ParentCoordVec}{\Vec{\ParentCoord}} \newcommand{\LocalSpace}{\SpaceHilbert{\SymbolLocalBasis}} \newcommand{\LocalSpaceDetailed}[2]{\Detailed{\LocalSpace}{#1}{#2}} \newcommand{\LocalSpaceOnMesh}{\ChildOfParent{\LocalSpace}{\Mesh}} \newcommand{\LocalBasisFunc}{\SymbolLocalBasis} \newcommand{\LocalBasisFuncDetailed}[2]{\Detailed{\LocalBasisFunc}{#1}{#2}} \newcommand{\LocalBasisFuncVec}{\Vec{\SymbolLocalBasis}} \newcommand{\LocalBasisFuncVecOnQuantity}[1]{\ChildOfParent{\LocalBasisFuncVec}{#1}} \newcommand{\LocalBasisFuncVecDetailed}[2]{\Detailed{\LocalBasisFuncVec}{#1}{#2}} \newcommand{\Mesh}{\boldsymbol{\textsf{M}}} \newcommand{\MeshAdmissible}{\ChildOfParent{\Mesh}{\operatorname{adm}}} \newcommand{\MeshSize}{h} \newcommand{\Elem}{\Set{E}} \newcommand{\Interface}{\Set{I}} \newcommand{\Face}{\Set{f}} \newcommand{\Edge}{\Set{e}} \newcommand{\Vertex}{\Set{v}} \newcommand{\CellArbitrary}[1]{\Set{#1}} \newcommand{\CellArbitraryOfDim}[2]{\ChildOfParent{\CellArbitrary{#1}}{#2}} \newcommand{\Cell}{\CellArbitrary{c}} \newcommand{\CellOfType}[1]{\ChildOfParent{\Cell}{#1}} \newcommand{\CellSet}{\Set{C}} \newcommand{\CellSetOfType}[1]{\ChildOfParent{\CellSet}{#1}} \newcommand{\CellFromQuantity}[1]{\Of{\operatorname{cell}}{#1}} \newcommand{\lbi}{\BasisFuncId} \newcommand{\lbii}{\Set{i}} \newcommand{\lbj}{\BasisFuncIdAlt} \newcommand{\lbjj}{\Set{j}} \newcommand{\lbkk}{\Set{k}} \newcommand{\IndexSet}{\Set{ID}} \newcommand{\IndexSetOf}[1]{\Of{\Set{ID}}{#1}} \newcommand{\IndexSetSet}{\boldsymbol{\IndexSet}} \newcommand{\IndexSetOnMesh}{\ChildOfParent{\IndexSet}{\Mesh}} \newcommand{\IndexOnQuantity}[2]{\ParentOfChild{#1}{#2}} \newcommand{\IndexOnQuantityIJ}[3]{\ParentOfChild{#1}{\TupleRoster{#2,#3}}} \newcommand{\LocalDegree}{\SymbolParentModifier{\PolynomialDegree}} \newcommand{\LocalDegreeAlt}{\SymbolParentModifier{q}} \newcommand{\LocalDegreeAltAlt}{\SymbolParentModifier{k}} \newcommand{\LocalDegreeInS}{\LocalDegree_{0}} \newcommand{\LocalDegreeInT}{\LocalDegree_{1}} \newcommand{\LocalDegreeDetailed}[2]{\Detailed{\bar{\PolynomialDegree}}{#1}{#2}} \newcommand{\LocalDegreeMinOf}[1]{\Of{\LocalDegree_{\min}}{#1}} \newcommand{\LocalDegreeTuple}{\Tuple{\LocalDegree}} \newcommand{\LocalDegreeElevMatComp}{\SymbolParentModifier{D}} \newcommand{\LocalDegreeElevMat}{\Mat{\LocalDegreeElevMatComp}} \newcommand{\LocalDegreeChange}{\Delta(\LocalDegree)} \newcommand{\LocalDegreeMinParallelPerp}[2]{\Of{\LocalDegree^{\parallel,\perp}_{\min}}{#1,#2}} \newcommand{\LocalDegreeMaxParallelPerp}[2]{\Of{\LocalDegree^{\parallel,\perp}_{\max}}{#1,#2}} \newcommand{\GlobalDegree}{\SymbolParamModifier{\PolynomialDegree}} \newcommand{\GlobalDegreeAlt}{\SymbolParamModifier{q}} \newcommand{\GlobalDegreeAltAlt}{\SymbolParamModifier{k}} \newcommand{\LocalDegreeParallel}{\LocalDegree^{\parallel}} \newcommand{\LocalDegreeParallelTo}[2]{\Of{\LocalDegreeParallel_{#1}}{#2}} \newcommand{\LocalDegreeTupleParallelTo}[2]{\Of{\LocalDegreeTuple^{\parallel}_{#1}}{#2}} \newcommand{\LocalDegreeMaxParallelTo}[1]{\Of{\LocalDegreeParallel_{\max}}{#1}} \newcommand{\LocalDegreeMinParallelTo}[1]{\Of{\LocalDegreeParallel_{\min}}{#1}} \newcommand{\LocalDegreePerp}{\LocalDegree^{\perp}} \newcommand{\LocalDegreePerpTo}[2]{\Of{\LocalDegreePerp_{#1}}{#2}} \newcommand{\LocalDegreeMaxPerpTo}[1]{\Of{\LocalDegreePerp_{\max}}{#1}} \newcommand{\LocalDegreeMinPerpTo}[1]{\Of{\LocalDegreePerp_{\min}}{#1}} \newcommand{\Smoothness}{\vartheta} \newcommand{\SmoothnessDetailed}[2]{\Detailed{\Smoothness}{#1}{#2}} \newcommand{\SmoothnessSet}{\boldsymbol{\Smoothness}} \newcommand{\SmoothnessMaxPerpTo}[1]{\Of{\SmoothnessDetailed{\max}{\perp}}{#1}} \newcommand{\SmoothnessVar}{\rho} \newcommand{\AdjacentCellSet}[1]{\ChildOfParent{\Set{ADJ}}{#1}} \newcommand{\AdjacentCellSetOf}[2]{\Of{\AdjacentCellSet{#1}}{#2}} \newcommand{\AdjacentMinCellSet}{\AdjacentCellSet{\min}} \newcommand{\AdjacentMinCellSetOf}[1]{\Of{\AdjacentMinCellSet}{#1}} \newcommand{\AdjacentMaxCellSet}{\AdjacentCellSet{\max}} \newcommand{\AdjacentMaxCellSetOf}[1]{\Of{\AdjacentMaxCellSet}{#1}} \newcommand{\AdjacentRing}[1]{\ChildOfParent{\Set{RING}}{#1}} \newcommand{\AdjacentRingOf}[2]{\Of{\AdjacentRing{#1}}{#2}} \newcommand{\AdjacentCardinal}{\Set{CRD}} \newcommand{\AdjacentCardinalOf}[1]{\Of{\AdjacentCardinal}{#1}} \newcommand{\Submesh}{\boldsymbol{\textsf{K}}} \newcommand{\SubmeshDomain}{\From{\SymbolParamModifier{\Domain}}{\Submesh}} \newcommand{\SubmeshDomainBdry}{\From{\SymbolParamModifier{\DomainBdry}}{\Submesh}} \newcommand{\SubmeshDomainChart}{\Vec{\varphi}} \newcommand{\SubmeshDomainChartOfQuantity}[1]{\ChildOfParent{\SubmeshDomainChart}{#1}} \newcommand{\SubmeshDomainParameter}{\mathsf{\alpha}} \newcommand{\SubmeshDomainParameterVec}{\Vec{\SubmeshDomainParameter}} \newcommand{\ParamDim}{{\SymbolParamModifier{\SymbolDim}}} \newcommand{\ParamDomain}{\SymbolParamModifier{\Domain}} \newcommand{\ParamDomainFrom}[1]{\From{\ParamDomain}{#1}} \newcommand{\ParamDomainBdry}{\SymbolParamModifier{\DomainBdry}} \newcommand{\ParamDomainBdryFrom}[1]{\From{\ParamDomainBdry}{#1}} \newcommand{\ParamDomainOnCell}{\ParentOfChild{\Cell}{\ParamDomain}} \newcommand{\ParamDomainOnMesh}{\ParentOfChild{\Mesh}{\ParamDomain}} \newcommand{\ParamDomainOnMeshAdmissible}{\ParentOfChild{\MeshAdmissible}{\ParamDomain}} \newcommand{\ParamDomainChart}[1]{{\ChildOfParent{\Vec{\phi}}{#1}}} \newcommand{\ParamDomainChartOf}[2]{\Of{\ParamDomainChart{#1}}{#2}} \newcommand{\ParamCoordComp}{s} \newcommand{\ParamCoordCompOnQuantity}[1]{\ParentOfChild{#1}{\ParamCoordComp}} \newcommand{\ParamCoordVec}{\Vec{s}} \newcommand{\ParamCoordVecOnQuantity}[1]{\ParentOfChild{#1}{\Vec{\ParamCoordComp}}} \newcommand{\ParamCoordVecPerpTo}[2]{\Of{\Detailed{\ParamCoordVec}{#1}{\perp}}{#2}} \newcommand{\ParamCoordVecParallelTo}[2]{\Of{\Detailed{\ParamCoordVec}{#1}{\parallel}}{#2}} \newcommand{\ParamLength}{\ell} \newcommand{\ParamLengthDetailed}[2]{\Detailed{\ParamLength}{#1}{#2}} \newcommand{\ParamBarycentric}[1]{\lambda_{#1}} \newcommand{\SpaceGlobal}{\SpaceHilbert{\SymbolGlobalBasis}} \newcommand{\SpaceGlobalOnMesh}{\ParentOfChild{\Mesh}{\SpaceGlobal}} \newcommand{\SpaceGlobalOnMeshAdmissible}{\ParentOfChild{\MeshAdmissible}{\SpaceGlobal}} \newcommand{\GlobalBasisFunc}{\SymbolGlobalBasis} \newcommand{\GlobalBasisFuncDetailed}[2]{\GlobalBasisFunc^{#1}_{#2}} \newcommand{\GlobalBasisFuncOnMesh}[1]{\GlobalBasisFuncDetailed{\Mesh}{#1}} \newcommand{\GlobalBasisFuncOnMeshAdmissible}[1]{\GlobalBasisFuncDetailed{\MeshAdmissible}{#1}} \newcommand{\GlobalBasisFuncSet}{\Set{\SymbolGlobalBasis}} \newcommand{\GlobalBasisFuncSetOnQuantity}[1]{\ChildOfParent{\GlobalBasisFuncSet}{#1}} \newcommand{\GlobalBasisFuncSetOnMesh}{\GlobalBasisFuncSetOnQuantity{\Mesh}} \newcommand{\GlobalBasisFuncSetOnMeshAdmissible}{\GlobalBasisFuncSetOnQuantity{\MeshAdmissible}} \newcommand{\GlobalBasisFuncVec}{\Vec{\SymbolGlobalBasis}} \newcommand{\GlobalBasisFuncVecOnQuantity}[1]{\ChildOfParent{\GlobalBasisFuncVec}{#1}} \newcommand{\GlobalBasisFuncVecOnMesh}{\GlobalBasisFuncVecOnQuantity{\Mesh}} \newcommand{\GlobalBasisFuncVecOnMeshAdmissible}{\GlobalBasisFuncVecOnQuantity{\MeshAdmissible}} \newcommand{\GlobalBasisFuncId}{A} \newcommand{\GlobalBasisFuncIdAlt}{B} \newcommand{\GlobalBasisFuncIdLocal}{a} \newcommand{\BasisFuncCoeff}{\SymbolCoeff} \newcommand{\BasisFuncCoeffsSet}{\Set{\BasisFuncCoeff}} \newcommand{\BasisFuncCoeffsSetFromMesh}{\From{\BasisFuncCoeffsSet}{\Mesh}} \newcommand{\BasisFuncCoeffsVec}{\Vec{\BasisFuncCoeff}} \newcommand{\BasisFuncCoeffsVecFromMesh}{\From{\BasisFuncCoeffsVec}{\Mesh}} \newcommand{\ExtractOp}[1]{\ParentOfChild{#1}{\Mat{C}}} \newcommand{\ExtractApproxOp}[1]{\ParentOfChild{#1}{\widetilde{\Mat{C}}}} \newcommand{\ReconstructOp}[1]{\ParentOfChild{#1}{\Mat{R}}} \newcommand{\ExtensionOp}[1]{\Mat{E}^{#1}} \newcommand{\ExtensionTolerance}{\ChildOfParent{\Tolerance}{\operatorname{ext}}} \newcommand{\BaseTopoLine}{\Set{L}} \newcommand{\BaseTopoLoop}{\Set{P}} \newcommand{\BaseTopoOneDGeneric}{\Mesh^{\textsf{1D}}} \newcommand{\BaseTopoGrid}{\Set{G}} \newcommand{\BaseTopoAnnulus}{\Set{A}} \newcommand{\BaseTopoTorus}{\Set{T}} \newcommand{\BaseTopoTriangle}{\Set{\Delta}} \newcommand{\BaseTopoMixed}{\Set{M}} \newcommand{\BaseTopoMultiPatch}{\Set{X}} \newcommand{\BaseTopoPatch}{\Set{H}} \newcommand{\DimId}{k} \newcommand{\DimIdOne}{\DimId_0} \newcommand{\DimIdTwo}{\DimId_1} \newcommand{\DimIdThree}{\DimId_2} \newcommand{\DimIdAlt}{n} \newcommand{\DimIdSet}{\Set{\DimId}} \newcommand{\NumLocalBasisFuncsOnElem}{\Size{\LocalBasisFuncVecOnQuantity{\Elem}}} \newcommand{\NumLocalBasisFuncsOnMesh}{\Size{\LocalBasisFuncVecOnQuantity{\Mesh}}} \newcommand{\NumLocalBasisFuncsOnMeshAdmissible}{\Size{\LocalBasisFuncVecOnQuantity{\MeshAdmissible}}} \newcommand{\NumGlobalBasisFuncsOnElem}{\Size{\GlobalBasisFuncVecOnQuantity{\Elem}}} \newcommand{\NumGlobalBasisFuncsOnMesh}{\Size{\GlobalBasisFuncVecOnMesh}} \newcommand{\NumGlobalBasisFuncsOnMeshAdmissible}{\Size{\GlobalBasisFuncVecOnMeshAdmissible}} \newcommand{\NumElemsInMesh}{\Size{\From{\CellSetOfType{\ParamDim}}{\Mesh}}} \newcommand{\StructuredSymbol}{\square} \newcommand{\StructuredSubmesh}[1]{\Submesh^{\StructuredSymbol}_{#1}} \newcommand{\ToGlobalDartOp}{\operatorname*{\mathsf{G}}} \newcommand{\ProlongationMat}{\Mat{P}} \newcommand{\RestrictionMat}{\Mat{R}}$$\newcommand{\SymbolGrevillePoint}{g} \newcommand{\GrevillePoint}{\Set{\SymbolGrevillePoint}} \newcommand{\GrevillePointDetailed}[2]{\Detailed{\GrevillePoint}{#1}{#2}} \newcommand{\GrevillePointOf}[1]{\Of{\GrevillePoint}{#1}} \newcommand{\GrevillePointOnQuantity}[1]{\ChildOfParent{\GrevillePoint}{#1}} \newcommand{\GrevillePointSet}{\Set{G}} \newcommand{\GrevillePointSetDetailed}[2]{\Detailed{\GrevillePointSet}{#1}{#2}} \newcommand{\GrevillePointSetOf}[1]{\Of{\GrevillePointSet}{#1}} \newcommand{\GrevillePointSetOnQuantity}[1]{\ChildOfParent{\GrevillePointSet}{#1}} \newcommand{\GrevillePointSetSet}{\boldsymbol{\GrevillePointSet}} \newcommand{\GrevillePointSetSetDetailed}[2]{\Detailed{\GrevillePointSetSet}{#1}{#2}} \newcommand{\GrevillePointSetSetOf}[1]{\Of{\GrevillePointSetSet}{#1}} \newcommand{\GrevillePointSetSetOnQuantity}[1]{\ChildOfParent{\GrevillePointSetSet}{#1}} \newcommand{\ParentGrevillePoint}[1]{\ParentOfChild{#1}{\Set{\SymbolParentModifier{\SymbolGrevillePoint}}}} \newcommand{\ParentGrevillePointOf}[1]{\Of{\Set{\SymbolParentModifier{\SymbolGrevillePoint}}}{#1}} \newcommand{\ConstraintCoeff}{r} \newcommand{\ConstraintCoeffsVec}{\Vec{\ConstraintCoeff}} \newcommand{\ConstraintCoeffsSet}{\boldsymbol{\Set{\ConstraintCoeff}}} \newcommand{\Constraint}{R} \newcommand{\ConstraintSet}{\Set{\Constraint}} \newcommand{\ConstraintSetOf}[1]{\Of{\ConstraintSet}{#1}} \newcommand{\ConstraintSetOnMesh}{\ConstraintSetOf{\Mesh}} \newcommand{\ConstraintMat}{\Mat{\Constraint}} \newcommand{\ConstraintMatOf}[1]{\Of{\ConstraintMat}{#1}} \newcommand{\ConstraintMatOnMesh}{\ConstraintMatOf{\Mesh}} \newcommand{\AlignedSet}{\Set{ALIGN}} \newcommand{\AlignedSetSet}{\boldsymbol{\AlignedSet}} \newcommand{\Spoke}{\rho} \newcommand{\InclDist}{\operatorname{inc}} \newcommand{\InclDistSet}{\Set{INC}} \newcommand{\ElemInterfPair}{\Set{t}} \newcommand{\SpaceNull}[1]{\Of{\SpaceHilbert{NV}}{#1}} \newcommand{\SpaceNullOnMesh}{\SpaceNull{\Mesh}} \newcommand{\NullVector}{\Vec{v}} \newcommand{\NullVectorOf}[1]{\Of{\NullVector}{#1}} \newcommand{\NullVectorArbitrary}[1]{\Vec{#1}} \newcommand{\NullVectorSet}{\boldsymbol{\textsf{NV}}} \newcommand{\NullVectorSetDetailed}[2]{\Detailed{\NullVectorSet}{#1}{#2}} \newcommand{\NullVectorSetOf}[1]{\Of{\NullVectorSet}{#1}} \newcommand{\NullVectorSetOnMesh}{\NullVectorSetOf{\Mesh}} \newcommand{\NullVectorSetOnMeshAdmissible}{\NullVectorSetOf{\MeshAdmissible}} \newcommand{\NullVectorBdrySet}[1]{\NullVectorSetDetailed{#1}{\operatorname{bdry}}} \newcommand{\NullVectorBdrySetOf}[2]{\Of{\NullVectorBdrySet{#1}}{#2}} \newcommand{\NullVectorMasterSet}{\NullVectorSet^{\operatorname{master}}} \newcommand{\NullVectorMasterSetDetailed}[1]{\NullVectorMasterSet_{#1}} \newcommand{\NullVectorMasterSetOf}[1]{\Of{\NullVectorMasterSet}{#1}} \newcommand{\NullVectorSubordinateSet}[1]{\NullVectorSetDetailed{#1}{\operatorname{sub}}} \newcommand{\NullVectorSubordinateSetOf}[2]{\Of{\NullVectorSubordinateSet{#1}}{#2}} \newcommand{\NullVectorSimpleSet}{\NullVectorSet^{\prime}} \newcommand{\NullVectorCompositeSet}{\NullVectorSet^{\prime\prime}} \newcommand{\UsplineNullVectorCoeff}{u} \newcommand{\UsplineNullVector}{\Vec{\UsplineNullVectorCoeff}} \newcommand{\NullVectorExpansionSet}{\ChildOfParent{\NullVectorSet}{\operatorname{X}}} \newcommand{\NullVectorExpansionSetOf}[2]{\Of{\NullVectorExpansionSet}{#1,#2}} \newcommand{\Graph}{G} \newcommand{\GraphCore}{K} \newcommand{\GraphDirectedEdge}{\Edge_{i,j}} \newcommand{\GraphDirectedEdgeIJ}[2]{\Edge_{#1,#2}} \newcommand{\GraphCoreToVertex}[1]{\Of{\Vertex}{#1}} \newcommand{\GraphInteractingEdges}{\Set{IE}} \newcommand{\GraphCoveredEdges}{\Set{CE}} \newcommand{\GraphFailedEdges}{\Set{FE}} \newcommand{\GraphCandidateEdges}{\Set{XE}} \newcommand{\GlobalBasisFuncNormalized}{\bar{\SymbolGlobalBasis}} \newcommand{\UsplineClass}[1]{\boldsymbol{\mathcal{\SymbolUspline}}^{#1}} \newcommand{\UsplineClassRegular}{R} \newcommand{\UsplineClassIrregular}{r} \newcommand{\UsplineClassContFinite}{H} \newcommand{\UsplineClassContInfinity}{h} \newcommand{\UsplineClassGlobalSmoothness}{K} \newcommand{\UsplineClassLocalSmoothness}{k} \newcommand{\UsplineClassGlobalDegree}{P} \newcommand{\UsplineClassLocalDegree}{p} \newcommand{\SuperscriptRHKP}{\UsplineClassRegular\UsplineClassContFinite\UsplineClassGlobalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptrHKP}{\UsplineClassIrregular\UsplineClassContFinite\UsplineClassGlobalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptRhKP}{\UsplineClassRegular\UsplineClassContInfinity\UsplineClassGlobalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptrhKP}{\UsplineClassIrregular\UsplineClassContInfinity\UsplineClassGlobalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptRHkP}{\UsplineClassRegular\UsplineClassContFinite\UsplineClassLocalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptrHkP}{\UsplineClassIrregular\UsplineClassContFinite\UsplineClassLocalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptRhkP}{\UsplineClassRegular\UsplineClassContInfinity\UsplineClassLocalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptrhkP}{\UsplineClassIrregular\UsplineClassContInfinity\UsplineClassLocalSmoothness\UsplineClassGlobalDegree} \newcommand{\SuperscriptRHKp}{\UsplineClassRegular\UsplineClassContFinite\UsplineClassGlobalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptrHKp}{\UsplineClassIrregular\UsplineClassContFinite\UsplineClassGlobalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptRhKp}{\UsplineClassRegular\UsplineClassContInfinity\UsplineClassGlobalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptrhKp}{\UsplineClassIrregular\UsplineClassContInfinity\UsplineClassGlobalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptRHkp}{\UsplineClassRegular\UsplineClassContFinite\UsplineClassLocalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptrHkp}{\UsplineClassIrregular\UsplineClassContFinite\UsplineClassLocalSmoothness\UsplineClassLocalDegree} \newcommand{\SuperscriptRhkp}{\UsplineClassRegular\UsplineClassContInfinity\UsplineClassLocalSmoothness\UsplineClassLocalDegree} \newcommand{\Superscriptrhkp}{\UsplineClassIrregular\UsplineClassContInfinity\UsplineClassLocalSmoothness\UsplineClassLocalDegree} \newcommand{\UsplineClassRHKP}{\UsplineClass{\SuperscriptRHKP}} \newcommand{\UsplineClassrHKP}{\UsplineClass{\SuperscriptrHKP}} \newcommand{\UsplineClassRhKP}{\UsplineClass{\SuperscriptRhKP}} \newcommand{\UsplineClassrhKP}{\UsplineClass{\SuperscriptrhKP}} \newcommand{\UsplineClassRHkP}{\UsplineClass{\SuperscriptRHkP}} \newcommand{\UsplineClassrHkP}{\UsplineClass{\SuperscriptrHkP}} \newcommand{\UsplineClassRhkP}{\UsplineClass{\SuperscriptRhkP}} \newcommand{\UsplineClassrhkP}{\UsplineClass{\SuperscriptrhkP}} \newcommand{\UsplineClassRHKp}{\UsplineClass{\SuperscriptRHKp}} \newcommand{\UsplineClassrHKp}{\UsplineClass{\SuperscriptrHKp}} \newcommand{\UsplineClassRhKp}{\UsplineClass{\SuperscriptRhKp}} \newcommand{\UsplineClassrhKp}{\UsplineClass{\SuperscriptrhKp}} \newcommand{\UsplineClassRHkp}{\UsplineClass{\SuperscriptRHkp}} \newcommand{\UsplineClassrHkp}{\UsplineClass{\SuperscriptrHkp}} \newcommand{\UsplineClassRhkp}{\UsplineClass{\SuperscriptRhkp}} \newcommand{\UsplineClassrhkp}{\UsplineClass{\Superscriptrhkp}} \newcommand{\Ribbon}{\Set{r}} \newcommand{\RibbonContinuityTransition}{\ChildOfParent{\Set{c}}{\Ribbon}} \newcommand{\RibbonDegreeTransition}{\ChildOfParent{\Set{d}}{\Ribbon}} \newcommand{\RibbonHead}{\ChildOfParent{\operatorname{head}}{\Ribbon}} \newcommand{\RibbonTail}{\ChildOfParent{\operatorname{tail}}{\Ribbon}} \newcommand{\RibbonOrigin}{\ChildOfParent{\operatorname{origin}}{\Ribbon}}$$\newcommand{\SymbolTime}{t} \newcommand{\SymbolClosestPointMap}{\kappa} \newcommand{\SymbolManifold}{m} \newcommand{\SymbolVolume}{v} \newcommand{\SymbolVolumeUpper}{V} \newcommand{\SymbolSurface}{s} \newcommand{\SymbolSurfaceUpper}{S} \newcommand{\SymbolCurve}{c} \newcommand{\SymbolCurveUpper}{C} \newcommand{\SymbolPoint}{p} \newcommand{\SymbolPointUpper}{P} \newcommand{\SymbolSegment}{\epsilon} \newcommand{\SymbolSpatialCoord}{x} \newcommand{\SymbolSegmentSet}{\boldsymbol{S}} \newcommand{\SymbolInside}{\bullet} \newcommand{\SymbolOutside}{\llcorner} \newcommand{\SymbolCut}{\ominus} \newcommand{\SymbolPartition}{P} \newcommand{\SymbolTessellation}{\boldsymbol{\vartriangle}} \newcommand{\SymbolQuadratureWeight}{w} \newcommand{\SymbolQuadratureSet}{Q} \newcommand{\SymbolQuadraturePoint}{q} \newcommand{\SymbolIntegrationPoint}{i} \newcommand{\SymbolManifoldCoeff}{x} \newcommand{\SymbolManifoldCoeffUpper}{X} \newcommand{\SymbolCADModified}{\boldsymbol{\mathfrak{c}}} \newcommand{\SymbolCAD}{\bar{\SymbolCADModified}} \newcommand{\SymbolCovariantBasisSurf}{a} \newcommand{\SymbolCovariantBasisVol}{g} \newcommand{\SymbolMetric}{m} \newcommand{\SymbolCurvature}{k} \newcommand{\SymbolLinearInterpolation}{\mathscr{l}} \newcommand{\SpatialDim}{\SymbolDim} \newcommand{\SpatialDomain}{\Reals^{\SymbolDim}} \newcommand{\SpatialCoord}{\SymbolSpatialCoord} \newcommand{\SpatialCoordX}{\SymbolSpatialCoord} \newcommand{\SpatialCoordY}{y} \newcommand{\SpatialCoordZ}{z} \newcommand{\SpatialCoordVec}{\Vec{\SpatialCoord}} \newcommand{\Manifold}{\Set{\SymbolManifold}} \newcommand{\ManifoldFrom}[1]{\From{\Manifold}{#1}} \newcommand{\Volume}{\Set{\SymbolVolume}} \newcommand{\VolumeFrom}[1]{\From{\Volume}{#1}} \newcommand{\Surface}{\Set{\SymbolSurface}} \newcommand{\SurfaceFrom}[1]{\From{\Surface}{#1}} \newcommand{\Curve}{\Set{\SymbolCurve}} \newcommand{\CurveFrom}[1]{\From{\Curve}{#1}} \newcommand{\Point}{\Set{\SymbolPoint}} \newcommand{\PointFrom}[1]{\From{\Point}{#1}} \newcommand{\ManifoldFromCAD}{\ManifoldFrom{\SymbolCAD}} \newcommand{\ManifoldFromCADModified}{\ManifoldFrom{\SymbolCADModified}} \newcommand{\ManifoldFromPartition}{\ManifoldFrom{\Partition}} \newcommand{\ManifoldFromMesh}{\ManifoldFrom{\Mesh}} \newcommand{\ManifoldFromMeshInterior}{\From{\Manifold^{\SymbolInside}}{\Mesh}} \newcommand{\ManifoldFromMeshAdmissible}{\ManifoldFrom{\MeshAdmissible}} \newcommand{\dVolume}{\Differential[\Volume]} \newcommand{\SurfaceFromCAD}{\SurfaceFrom{\SymbolCAD}} \newcommand{\SurfaceFromCADModified}{\SurfaceFrom{\SymbolCADModified}} \newcommand{\SurfaceFromWildCard}{\Surface^{\wildcard}} \newcommand{\dSurface}{\Differential[\Surface]} \newcommand{\Segment}{\SymbolSegment} \newcommand{\SegmentOf}[1]{\ParentOfChild{#1}{\Segment}} \newcommand{\SegmentOfVolume}{\SegmentOf{\Volume}} \newcommand{\SegmentOfSurface}{\SegmentOf{\Surface}} \newcommand{\Hex}{\Segment_{\operatorname{hex}}} \newcommand{\Tet}{\Segment_{\operatorname{tet}}} \newcommand{\SegmentSet}{\Set{\SymbolSegmentSet}} \newcommand{\SegmentSetOf}[1]{\Of{\SegmentSet}{#1}} \newcommand{\SegmentSetOutside}{\SegmentSet^{\SymbolOutside}} \newcommand{\SegmentSetInside}{\SegmentSet^{\SymbolInside}} \newcommand{\SegmentSetHybrid}{\SegmentSet^{\SymbolCut}} \newcommand{\Partition}{\Set{\SymbolPartition}} \newcommand{\PartitionOf}[1]{\Of{\Partition}{#1}} \newcommand{\Tessellation}{\SymbolTessellation} \newcommand{\TessellationOf}[1]{\Of{\Tessellation}{#1}} \newcommand{\dTessellation}{\Differential[\Tessellation]} \newcommand{\ManifoldCoeff}{\SymbolManifoldCoeff} \newcommand{\ManifoldCoeffFromMesh}{\From{\SymbolManifoldCoeff}{\Mesh}} \newcommand{\ManifoldCoeffVec}{\Vec{\SymbolManifoldCoeff}} \newcommand{\ManifoldCoeffVecFromMesh}{\From{\Vec{\SymbolManifoldCoeff}}{\Mesh}} \newcommand{\ManifoldCoeffsVec}{\Vec{\SymbolManifoldCoeffUpper}} \newcommand{\ManifoldCoeffsVecFromMesh}{\From{\Vec{\SymbolManifoldCoeffUpper}}{\Mesh}} \newcommand{\ManifoldCoeffsVecFromMeshCell}{\From{\Vec{\SymbolManifoldCoeffUpper}}{\ParentOfChild{\Mesh}{\Cell}}} \newcommand{\ManifoldCoeffsVecFromSubmeshCell}{\From{\Vec{\SymbolManifoldCoeffUpper}}{\ParentOfChild{\Submesh}{\Cell}}} \newcommand{\ManifoldCoeffsSet}{\Set{\SymbolManifoldCoeffUpper}} \newcommand{\ManifoldCoeffsSetFromMesh}{\From{\Set{\SymbolManifoldCoeffUpper}}{\Mesh}} \newcommand{\ManifoldCoeffsSetFromMeshCell}{\From{\Set{\SymbolManifoldCoeffUpper}}{\ParentOfChild{\Mesh}{\Cell}}} \newcommand{\ManifoldCoeffsSetFromSubmeshCell}{\From{\Set{\SymbolManifoldCoeffUpper}}{\ParentOfChild{\Submesh}{\Cell}}} \newcommand{\ManifoldMap}[2]{\From{#2}{#1}} \newcommand{\ManifoldTemporalMap}[2]{\From{#2}{#1,\SymbolTime}} \newcommand{\ManifoldFromMeshMap}{\ManifoldMap{\ParamDomainOnMesh}{\Manifold}} \newcommand{\ManifoldFromPartitionMap}{\ManifoldMap{\ChildOfParent{\ParamDomain}{\Partition}}{\Manifold}} \newcommand{\VolumeMap}{\ManifoldMap{\ParamDomain}{\Volume}} \newcommand{\SurfaceMap}{\ManifoldMap{\ParamDomainBdry}{\Surface}} \newcommand{\ManifoldMapDef}[2]{\FuncDef{\ManifoldMap{#1}{#2}}{#1}{#2}} \newcommand{\ManifoldTemporalMapDef}[2]{\FuncDef{\ManifoldTemporalMap{#1}{#2}}{#1\otimes\Reals_{+}}{#2}} \newcommand{\ManifoldFromMeshMapDef}{\ManifoldMapDef{\ParamDomainOnMesh}{\Manifold}} \newcommand{\ManifoldFromPartitionMapDef}{\ManifoldMapDef{\ChildOfParent{\ParamDomain}{\Partition}}{\Manifold}} \newcommand{\VolumeMapDef}{\ManifoldMapDef{\ParamDomain}{\Volume}} \newcommand{\SurfaceMapDef}{\ManifoldMapDef{\ParamDomainBdry}{\Surface}} \newcommand{\ManifoldCoord}{x} \newcommand{\ManifoldCoordDetailed}[2]{\ParentOfChild{#1}{#2}} \newcommand{\ManifoldCoordVec}{\Vec{x}} \newcommand{\ManifoldCoordVecDetailed}[2]{\ParentOfChild{#1}{\Vec{#2}}} \newcommand{\CovariantBasisSurfVec}{\Vec{\SymbolCovariantBasisSurf}} \newcommand{\CovariantBasisVolVec}{\Vec{\SymbolCovariantBasisVol}} \newcommand{\MetricComp}{\SymbolMetric} \newcommand{\MetricTen}{\Mat{\MetricComp}} \newcommand{\CurvatureComp}{\SymbolCurvature} \newcommand{\CurvatureTen}{\Mat{\CurvatureComp}} \newcommand{\QuadraturePoint}[2]{\SymbolQuadraturePoint_{#1}^{#2}} \newcommand{\qi}{\SymbolQuadraturePoint^{\SymbolInside}} \newcommand{\qo}{\SymbolQuadraturePoint^{\SymbolOutside}} \newcommand{\qoa}{\QuadraturePoint{\lbi}{\SymbolOutside}} \newcommand{\QuadraturePointToElemIndexMap}[1]{\operatorname{qe}(#1)} \newcommand{\QuadratureWeight}[2]{\SymbolQuadratureWeight_{#1}^{#2}} \newcommand{\QuadratureWeightUni}[1]{\SymbolQuadratureWeight_{\SymbolQuadraturePoint_{#1}}} \newcommand{\wi}{\SymbolQuadratureWeight^{\SymbolInside}} \newcommand{\wo}{\SymbolQuadratureWeight^{\SymbolOutside}} \newcommand{\woa}{\QuadratureWeight{\lbi}{\SymbolOutside}} \newcommand{\QuadratureSet}{\Set{\SymbolQuadratureSet}} \newcommand{\QuadratureSetOf}[1]{\Of{\QuadratureSet}{#1}} \newcommand{\QuadratureSetInside}{\QuadratureSet^{\SymbolInside}} \newcommand{\QuadratureSetOutside}{\QuadratureSet^{\SymbolOutside}} \newcommand{\ClosestPointMap}[1]{\ParentOfChild{#1}{\Vec{\SymbolClosestPointMap}}} \newcommand{\ClosestPointMapCAD}{\ClosestPointMap{\SymbolCAD}} \newcommand{\ClosestPointMapCADModified}{\ClosestPointMap{\SymbolCADModified}} \newcommand{\ProjectionInto}[1]{\From{\Projection}{#1}} \newcommand{\ProjectionIntoCell}{\ProjectionInto{\Cell}} \newcommand{\ProjectionIntoElem}{\ProjectionInto{\Elem}} \newcommand{\ProjectionIntoMesh}{\ProjectionInto{\Mesh}} \newcommand{\ProjectionIntoMeshAdmissible}{\ProjectionInto{\MeshAdmissible}} \newcommand{\LinearInterpolationOp}[1]{\From{\SymbolLinearInterpolation}{#1}} \newcommand{\BoundingVolume}[1]{\Of{\Set{bv}}{#1}} \newcommand{\BoundingVolumeSet}[1]{\Of{\Set{BV}}{#1}} \newcommand{\BoundingVolumeTree}[1]{\Of{\Set{BVH}}{#1}} \newcommand{\AxisAlignedBoundingBox}[1]{\Of{\Set{AABB}}{#1}} \newcommand{\GapTolerance}{\ChildOfParent{\Tolerance}{\operatorname{gap}}} \newcommand{\PointInversionDistanceCost}[2]{d\left(#1,#2\right)} \newcommand{\InversionPointComp}{p} \newcommand{\InversionPointTen}{\Vec{p}} \newcommand{\ParametricConstraint}{C} \newcommand{\ParametricConstraintLMComp}{\LagrangeMultCurrComp} \newcommand{\ParametricConstraintLMTen}{\LagrangeMultCurrTen} \newcommand{\ParametricConstraintSlack}{\sigma} \newcommand{\ParametricConstraintSlackTen}{\Vec{\sigma}} \newcommand{\InverseMapTangentComp}{T} \newcommand{\LowerBound}[2]{B^l\left(#1,#2\right)} \newcommand{\UpperBound}[2]{B^u\left(#1,#2\right)} \newcommand{\UpperBoundValue}{U} \newcommand{\UpperBoundValuePoints}{U^{\text{point}}} \newcommand{\NearestCells}[3][]{\Set{C}_{#1}\left(#2,#3\right)} \newcommand{\BVHNearestGeom}[3][]{c_{#1}\left(#2,#3\right)} \newcommand{\CellSetCut}{\CellSetOfType{\SymbolCut}} \newcommand{\TrimmingSurface}{\Surface} \newcommand{\CutSetOf}[1]{\From{\SegmentSetHybrid}{#1}} \newcommand{\InteriorSetOf}[1]{\From{\SegmentSetInside}{#1}} \newcommand{\ExteriorSetOf}[1]{\From{\SegmentSetOutside}{#1}} \newcommand{\ParametricAtlas}{\From{\ParamDomain}{\Mesh}} \newcommand{\ParametricBdryAtlas}{\From{\ParamDomainBdry}{\Mesh}} \newcommand{\SkinAtlas}{\From{\ParamDomainBdry}{\VolumeAtlas,\TrimmingSurface}} \newcommand{\ManifoldAtlas}{\ManifoldFromMesh} \newcommand{\VolumeAtlas}{\VolumeFrom{\Mesh}} \newcommand{\TrimmedAtlas}{\From{\ParamDomain}{\VolumeAtlas,\TrimmingSurface}} \newcommand{\ParametricChart}{\ManifoldMap{\ParentOfChild{\Cell}{\ParentDomain}}{\ParamDomainChart{}}} \newcommand{\ManifoldChart}{\ManifoldMap{\ParentOfChild{\Cell}{\ParamDomain}}{\mathbf{\mathsf{\chi}}}} \newcommand{\InvManifoldChart}{\ManifoldChart^{-1}} \newcommand{\TrimmingChart}{\From{\ParamDomainChart{}}{\hat{\Face}}} \newcommand{\SkinChart}{\ManifoldMap{\ParentDomainBdryDetailed{}{\Cell}}{\ParamDomainChart{}}} \newcommand{\MeshTrimmed}{\boldsymbol{\textsf{T}}} \newcommand{\GlobalBasisFuncSetOnMeshTrimmed}{\Of{\Set{GF}}{\MeshTrimmed}} \newcommand{\NumGlobalBasisFuncsOnMeshTrimmed}{\Size{\GlobalBasisFuncSetOnMeshTrimmed}} \newcommand{\NumElemsInMeshTrimmed}{\Size{\MeshTrimmed}} \newcommand{\LocalSpaceOnMeshTrimmed}{\ParentOfChild{\MeshTrimmed}{\LocalSpace}}$$\newcommand{\SymbolDisp}{u} \newcommand{\SymbolDispUpper}{U} \newcommand{\SymbolVelocity}{v} \newcommand{\SymbolVelocityUpper}{V} \newcommand{\SymbolAcceleration}{a} \newcommand{\SymbolAccelerationUpper}{A} \newcommand{\SymbolMass}{M} \newcommand{\SymbolLength}{L} \newcommand{\SymbolArea}{A} \newcommand{\SymbolConstraint}{g} \newcommand{\SymbolConstraintUpper}{G} \newcommand{\SymbolPenalty}{\varepsilon} \newcommand{\SymbolLagrangeMultRef}{\Lambda} \newcommand{\SymbolLagrangeMultCurr}{\lambda} \newcommand{\SymbolFict}{f} \newcommand{\SymbolTracForce}{h} \newcommand{\SymbolTracForceUpper}{H} \newcommand{\SymbolBodyForce}{b} \newcommand{\SymbolBodyForceUpper}{B} \newcommand{\SymbolVolumeRef}{\SymbolVolumeUpper} \newcommand{\SymbolSurfaceRef}{\SymbolSurfaceUpper} \newcommand{\SymbolVolumeCurr}{\SymbolVolume} \newcommand{\SymbolSurfaceCurr}{\SymbolSurface} \newcommand{\SymbolDeformGradInc}{f} \newcommand{\SymbolDeformGrad}{F} \newcommand{\SymbolStrainGreenLagrange}{E} \newcommand{\SymbolDeformCauchyGreenLeft}{b} \newcommand{\SymbolDeformCauchyGreenRight}{C} \newcommand{\SymbolStrainDisp}{B} \newcommand{\SymbolStrainSymGrad}{\epsilon} \newcommand{\SymbolStrainPlasticEquiv}{\alpha} \newcommand{\SymbolHardeningPlastic}{k} \newcommand{\SymbolYieldFunction}{f} \newcommand{\SymbolConsistencyParameterPlastic}{\gamma} \newcommand{\SymbolPlasticModuliScaling}{\beta} \newcommand{\SymbolYieldSurfaceNormal}{n} \newcommand{\SymbolStrainPlasticFailure}{\alpha^{f}} \newcommand{\SymbolMaterialFailureIndicator}{D^{f}} \newcommand{\SymbolEnergy}{\Pi} \newcommand{\SymbolEnergyDensityStrain}{\Psi} \newcommand{\SymbolEnergyDensityStrainVol}{U} \newcommand{\SymbolResidual}{R} \newcommand{\SymbolForce}{F} \newcommand{\SymbolStiffness}{K} \newcommand{\SymbolSolution}{d} \newcommand{\ParamDomainBdryDisp}{\ParamDomainBdry^{\SymbolDisp}} \newcommand{\ParamDomainBdryTrac}{\ParamDomainBdry^{\SymbolTracForce}} \newcommand{\VolumeRef}{\Set{\SymbolVolumeRef}} \newcommand{\dVolumeRef}{\Differential[\VolumeRef]} \newcommand{\SurfaceRef}{\Set{\SymbolSurfaceRef}} \newcommand{\TessellationOfSurfaceRef}{\TessellationOf{\SurfaceRef}} \newcommand{\SurfaceRefEpsilonBall}{\SurfaceRef^{\epsilon}} \newcommand{\TessellationOfSurfaceRefEpsilonBall}{\TessellationOf{\SurfaceRefEpsilonBall}} \newcommand{\SurfaceRefWildCard}{\SurfaceRef^{\wildcard}} \newcommand{\SurfaceRefDisp}{\SurfaceRef^{\SymbolDisp}} \newcommand{\SurfaceRefTrac}{\SurfaceRef^{\SymbolTracForce}} \newcommand{\dSurfaceRef}{\Differential[\SurfaceRef]} \newcommand{\RefCoord}[1]{\ManifoldCoordDetailed{#1}{X}} \newcommand{\RefCoordVec}[1]{\ManifoldCoordVecDetailed{#1}{X}} \newcommand{\X}{\Vec{X}} \newcommand{\Y}{\Vec{Y}} \newcommand{\VolumeCurr}{\Set{\SymbolVolumeCurr}} \newcommand{\dVolumeCurr}{\Differential[\VolumeCurr]} \newcommand{\SurfaceCurr}{\Set{\SymbolSurfaceCurr}} \newcommand{\SurfaceCurrDisp}{\SurfaceCurr^{\SymbolDisp}} \newcommand{\SurfaceCurrTrac}{\SurfaceCurr^{\SymbolTracForce}} \newcommand{\dSurfaceCurr}{\Differential[\SurfaceCurr]} \newcommand{\CurrCoord}[1]{\ManifoldCoordDetailed{#1}{x}} \newcommand{\CurrCoordVec}[1]{\ManifoldCoordVecDetailed{#1}{x}} \newcommand{\VolumeRefMap}{\ManifoldMap{\ParamDomain}{\VolumeRef}} \newcommand{\SurfaceRefDispMap}{\ManifoldMap{\ParamDomainBdryDisp}{\SurfaceRefDisp}} \newcommand{\SurfaceRefTracMap}{\ManifoldMap{\ParamDomainBdryTrac}{\SurfaceRefTrac}} \newcommand{\VolumeRefMapDef}{\ManifoldMapDef{\ParamDomain}{\VolumeRef}} \newcommand{\SurfaceRefDispMapDef}{\ManifoldMapDef{\ParamDomainBdryDisp}{\SurfaceRefDisp}} \newcommand{\SurfaceRefTracMapDef}{\ManifoldMapDef{\ParamDomainBdryTrac}{\SurfaceRefTrac}} \newcommand{\GenericDeformMap}{\Vec{\varphi}} \newcommand{\GenericDeformMapDef}{\FuncDef{\GenericDeformMap}{\VolumeRef}{\VolumeCurr}} \newcommand{\VolumeDeformMap}{\ManifoldMap{\VolumeRef}{\VolumeCurr}} \newcommand{\VolumeDeformMapAtX}{\Of{\VolumeDeformMap}{\X}} \newcommand{\VolumeDeformMapAtY}{\Of{\VolumeDeformMap}{\Y}} \newcommand{\VolumeDeformTemporalMap}{\ManifoldTemporalMap{\VolumeRef}{\VolumeCurr}} \newcommand{\SurfaceDeformDispMap}{\ManifoldMap{\SurfaceRefDisp}{\SurfaceCurrDisp}} \newcommand{\SurfaceDeformTracMap}{\ManifoldMap{\SurfaceRefTrac}{\SurfaceCurrTrac}} \newcommand{\VolumeDeformMapDef}{\ManifoldMapDef{\VolumeRef}{\VolumeCurr}} \newcommand{\VolumeDeformTemporalMapDef}{\ManifoldTemporalMapDef{\VolumeRef}{\VolumeCurr}} \newcommand{\DispTrialSpaceOverVolumeRef}{\SpaceHilbert{U}(\VolumeRef)} \newcommand{\DispTrialSpaceOverVolumeCurr}{\SpaceHilbert{U}(\VolumeCurr)} \newcommand{\DispTrialFuncCurr}{\DispFieldCurrTen} \newcommand{\DispTestSpaceOverVolumeRef}{\SpaceHilbert{V}(\VolumeRef)} \newcommand{\DispTestSpaceOverVolumeCurr}{\SpaceHilbert{V}(\VolumeCurr)} \newcommand{\DispTestFuncCurr}{\Vec{v}} \newcommand{\PressureTrialSpaceOverVolumeCurr}{\SpaceHilbert{P}(\VolumeCurr)} \newcommand{\PressureTrialFuncCurr}{\Pressure} \newcommand{\PressureTestSpaceOverVolumeCurr}{\SpaceHilbert{Q}(\VolumeCurr)} \newcommand{\PressureTestFuncCurr}{q} \newcommand{\DispPressureTrialSpaceOverVolumeCurr}{\DispTrialSpaceOverVolumeCurr \otimes \PressureTrialSpaceOverVolumeCurr} \newcommand{\DispPressureTestSpaceOverVolumeCurr}{\DispTestSpaceOverVolumeCurr \otimes \PressureTestSpaceOverVolumeCurr} \newcommand{\DispFieldRefComp}{\SymbolDispUpper} \newcommand{\DispFieldRefTen}{\Vec{\DispFieldRefComp}} \newcommand{\DispFieldRefTenAtX}{\Of{\DispFieldRefTen}{\X}} \newcommand{\DispFieldRefTenAtY}{\Of{\DispFieldRefTen}{\Y}} \newcommand{\DispFieldRefTenVariation}{\delta\DispFieldRefTen} \newcommand{\DispFieldRefTenVariationAtX}{\Of{\DispFieldRefTenVariation}{\X}} \newcommand{\DispFieldRefTenVariationAtY}{\Of{\DispFieldRefTenVariation}{\Y}} \newcommand{\DispFieldCurrComp}{\SymbolDisp} \newcommand{\DispFieldCurrTen}{\Vec{\DispFieldCurrComp}} \newcommand{\VelFieldCurrTen}{\dot{\DispFieldCurrTen}} \newcommand{\AccFieldCurrTen}{\ddot{\DispFieldCurrTen}} \newcommand{\AccFieldCurrComp}{\ddot{\DispFieldCurrComp}} \newcommand{\PressureFieldCurr}{\Pressure} \newcommand{\DeformGradComp}{\SymbolDeformGrad} \newcommand{\DeformGradTen}{\Mat{\DeformGradComp}} \newcommand{\DeformGradDet}{\Det{\DeformGradTen}} \newcommand{\JacDetOfTessellation}{\ParentOfChild{{\SymbolPartition^T}}{\JacDet}} \newcommand{\KinematicElast}{e} \newcommand{\KinematicInelast}{i} \newcommand{\KinematicPlast}{p} \newcommand{\KinematicTherm}{\theta} \newcommand{\DeformGradIncComp}{\SymbolDeformGradInc} \newcommand{\DeformGradIncTen}{\Mat{\DeformGradIncComp}} \newcommand{\DeformGradIncDevComp}{\bar{\SymbolDeformGradInc}} \newcommand{\DeformGradIncDevTen}{\bar{\Mat{\DeformGradIncComp}}} \newcommand{\JAsDeformGradDet}{J} \newcommand{\StretchTen}{\Mat{V}} \newcommand{\StrainGreenLagrangeComp}{\SymbolStrainGreenLagrange} \newcommand{\StrainGreenLagrangeTen}{\Mat{\StrainGreenLagrangeComp}} \newcommand{\StrainGreenLagrangeVoigt}{\widetilde{\StrainGreenLagrangeTen}} \newcommand{\DeformCauchyGreenRightComp}{\SymbolDeformCauchyGreenRight} \newcommand{\DeformCauchyGreenRightTen}{\Mat{\DeformCauchyGreenRightComp}} \newcommand{\DeformCauchyGreenLeftComp}{\SymbolDeformCauchyGreenLeft} \newcommand{\DeformCauchyGreenLeftTen}{\Mat{\DeformCauchyGreenLeftComp}} \newcommand{\DeformCauchyGreenLeftDevComp}{\bar{\SymbolDeformCauchyGreenLeft}} \newcommand{\DeformCauchyGreenLeftDevTen}{\bar{\Mat{\DeformCauchyGreenLeftComp}}} \newcommand{\StrainSymGradComp}{\SymbolStrainSymGrad} \newcommand{\StrainSymGradTen}{\Mat{\StrainSymGradComp}} \newcommand{\StrainDispMat}{\Mat{\SymbolStrainDisp}} \newcommand{\DeformCauchyGreenRightDevComp}{\bar{\SymbolDeformCauchyGreenRight}} \newcommand{\DeformCauchyGreenRightDevTen}{\bar{\Mat{\DeformCauchyGreenRightComp}}} \newcommand{\KinematicElastoplast}{{\rm ep}} \newcommand{\Trial}{{\rm trial}} \newcommand{\DeformCauchyGreenLeftElasticTenDef}{\DeformCauchyGreenLeftTen^\KinematicElast \DefEq \DeformGradTen^\KinematicElast \DeformGradTen^{\KinematicElast T}} \newcommand{\DeformCauchyGreenLeftElasticDevTenDef}{{\DeformCauchyGreenLeftDevTen}^\KinematicElast \DefEq {\JAsDeformGradDet}^{\KinematicElast-\frac{2}{3}} {\DeformCauchyGreenLeftTen}^\KinematicElast} \newcommand{\DeformCauchyGreenRightPlasticTenDef}{\DeformCauchyGreenRightTen^\KinematicPlast \DefEq \DeformGradTen^{\KinematicPlast T} \DeformGradTen^\KinematicPlast} \newcommand{\StrainPlasticEquiv}{\SymbolStrainPlasticEquiv} \newcommand{\ConsistencyParameterPlasticDiscrete}{\Delta \SymbolConsistencyParameterPlastic} \newcommand{\YieldSurfaceNormalComp}{\SymbolYieldSurfaceNormal} \newcommand{\YieldSurfaceNormalTen}{\Mat{\SymbolYieldSurfaceNormal}} \newcommand{\StrainPlasticFailure}{\SymbolStrainPlasticFailure} \newcommand{\MaterialFailureIndicator}{\SymbolMaterialFailureIndicator} \newcommand{\DispFieldOverVolumeCurrTenDef}{\FuncDef{\DispFieldCurrTen}{\VolumeCurr}{\SpatialDomain}} \newcommand{\DispFieldOverVolumeCurrTemporalTenDef}{\FuncDef{\DispFieldCurrTen}{\VolumeCurr\otimes\Reals_{+}}{\SpatialDomain}} \newcommand{\DeformGradOverVolumeRefTenDef}{\FuncDef{\DeformGradTen}{\VolumeRef}{\VolumeCurr}} \newcommand{\StrainGreenLagrangeOverVolumeRefTenDef}{\FuncDef{\StrainGreenLagrangeTen}{\VolumeRef}{\VolumeCurr}} \newcommand{\DeformCauchyGreenLeftTenDef}{\DeformCauchyGreenLeftTen \DefEq \DeformGradTen \DeformGradTen^T} \newcommand{\DeformCauchyGreenLeftDevTenDef}{\DeformCauchyGreenLeftDevTen \DefEq \DeformGradDet^{-\frac{2}{3}} \DeformCauchyGreenLeftTen} \newcommand{\StressPKOneTen}{\Mat{P}} \newcommand{\StressPKTwoComp}{S} \newcommand{\StressPKTwoTen}{\Mat{\StressPKTwoComp}} \newcommand{\StressPKTwoVoigt}{\widetilde{\StressPKTwoTen}} \newcommand{\StressCauchyComp}{\sigma} \newcommand{\StressCauchyTen}{\Mat{\sigma}} \newcommand{\StressCauchyVoigt}{\widetilde{\StressCauchyTen}} \newcommand{\StressCauchyMean}{p} \newcommand{\StressCauchyYield}{\StressCauchyComp_Y} \newcommand{\StressKirchhoffComp}{\tau} \newcommand{\StressKirchhoffTen}{\Mat{\tau}} \newcommand{\StressKirchhoffVoigt}{\widetilde{\StressKirchhoffTen}} \newcommand{\StressKirchhoffDevComp}{s} \newcommand{\StressKirchhoffDevTen}{\Mat{\StressKirchhoffDevComp}} \newcommand{\StressVonMises}{\sigma^{VMS}} \newcommand{\ModuliElastRefComp}{C} \newcommand{\ModuliElastRefTen}{\Mat{\ModuliElastRefComp}} \newcommand{\ModuliElastRefVoigt}{\widetilde{\ModuliElastRefTen}} \newcommand{\ModuliElastCurrComp}{c} \newcommand{\ModuliElastCurrTen}{\Mat{\ModuliElastCurrComp}} \newcommand{\ModuliElastCurrVoigt}{\widetilde{\ModuliElastCurrTen}} \newcommand{\MaterialYoungsModulus}{E} \newcommand{\MaterialPoissonsRatio}{\nu} \newcommand{\MaterialMassDensity}{\rho} \newcommand{\MaterialBulkModulus}{\kappa} \newcommand{\MaterialShearModulus}{\mu} \newcommand{\MaterialTimescale}{\tau} \newcommand{\MaterialShearModulusDev}{\bar{\MaterialShearModulus}} \newcommand{\MaterialThermalExpansion}{\alpha_{\text{thermal}}} \newcommand{\Area}{\SymbolArea} \newcommand{\Length}{\SymbolLength} \newcommand{\Width}{b} \newcommand{\Height}{h} \newcommand{\DiameterElem}{h} \newcommand{\UnitNormalRefComp}{N} \newcommand{\UnitNormalRefTen}{\Vec{\UnitNormalRefComp}} \newcommand{\UnitNormalCurrComp}{\UnitNormalComp} \newcommand{\UnitNormalCurrTen}{\UnitNormal} \newcommand{\PenaltyDisp}{\SymbolPenalty_{\SymbolDisp}} \newcommand{\BodyForceRefComp}{\SymbolBodyForceUpper} \newcommand{\BodyForceRefTen}{\Vec{\BodyForceRefComp}} \newcommand{\BodyForceCurrComp}{\SymbolBodyForce} \newcommand{\BodyForceCurrTen}{\Vec{\BodyForceCurrComp}} \newcommand{\TracForceRefComp}{\SymbolTracForceUpper} \newcommand{\TracForceRefTen}{\Vec{\TracForceRefComp}} \newcommand{\TracForceCurrComp}{\SymbolTracForce} \newcommand{\TracForceCurrTen}{\Vec{\TracForceCurrComp}} \newcommand{\Pressure}{p} \newcommand{\BodyForceOverVolumeCurrTenDef}{\FuncDef{\BodyForceCurrTen}{\VolumeCurr}{\SpatialDomain}} \newcommand{\BodyForceOverVolumeRefTenDef}{\BodyForceRefTen \DefEq \BodyForceCurrTen \circ \VolumeDeformMap} \newcommand{\TracForceOverSurfaceCurrTracTenDef}{\FuncDef{\TracForceCurrTen}{\SurfaceCurrTrac}{\SpatialDomain}} \newcommand{\TracForceOverSurfaceRefTracTenDef}{\TracForceRefTen \DefEq \TracForceCurrTen \circ \SurfaceDeformTracMap} \newcommand{\ConstraintRefTen}{\Vec{\SymbolConstraintUpper}} \newcommand{\LagrangeMultTestSpaceOverSurfaceRef}{\delta \LagrangeMultTrialSpaceOverSurfaceRef} \newcommand{\LagrangeMultTrialSpaceOverSurfaceRef}{\SpaceHilbert{L}(\SurfaceRefDisp)} \newcommand{\LagrangeMultRefComp}{\SymbolLagrangeMultRef} \newcommand{\LagrangeMultRefTen}{\Vec{\LagrangeMultRefComp}} \newcommand{\LagrangeMultCurrComp}{\SymbolLagrangeMultCurr} \newcommand{\LagrangeMultCurrTen}{\Vec{\LagrangeMultCurrComp}} \newcommand{\Energy}{\SymbolEnergy} \newcommand{\EnergyElastic}{\Energy} \newcommand{\EnergyElasticOf}[1]{\Of{\EnergyElastic}{#1}} \newcommand{\EnergyConstraint}{\Energy^{\LagrangeMultCurrComp}} \newcommand{\EnergyDensityStrain}{\SymbolEnergyDensityStrain} \newcommand{\EnergyDensityStrainVol}{\SymbolEnergyDensityStrainVol} \newcommand{\EnergyDensityStrainDev}{\bar{\SymbolEnergyDensityStrain}} \newcommand{\ResidualComp}{\SymbolResidual} \newcommand{\ResidualVec}{\Vec{\ResidualComp}} \newcommand{\ForceVec}{\Vec{\SymbolForce}} \newcommand{\ForceInternalVec}{\ForceVec^{int}} \newcommand{\ForceExternalVec}{\ForceVec^{ext}} \newcommand{\ForceConstraintPenaltyVec}{\ForceVec^{\PenaltyDisp}} \newcommand{\ForceConstraintLagrangeVec}{\ForceVec^{\lambda1}} \newcommand{\ForceConstraintVec}{\ForceVec^{\lambda2}} \newcommand{\StiffnessMat}{\Mat{\SymbolStiffness}} \newcommand{\StiffnessMaterialMat}{\StiffnessMat^{m}} \newcommand{\StiffnessGeometricMat}{\StiffnessMat^{g}} \newcommand{\StiffnessPenaltyMat}{\StiffnessMat^{\PenaltyDisp}} \newcommand{\StiffnessLagrangeMat}{\StiffnessMat^{\LagrangeMultCurrComp}} \newcommand{\MassMat}{\Mat{\SymbolMass}} \newcommand{\LumpedMassMat}{\widetilde{\MassMat}} \newcommand{\SolutionComp}{\SymbolSolution} \newcommand{\Solution}{\Vec{\SymbolSolution}} \newcommand{\SolutionDisp}{\Solution} \newcommand{\SolutionVel}{\Vec{\SymbolVelocity}} \newcommand{\SolutionAcc}{\Vec{\SymbolAcceleration}} \newcommand{\SolutionLagrange}{\Solution^{\LagrangeMultCurrComp}} \newcommand{\Flex}[1]{\mathcal{F}_{#1}} \newcommand{\FlexIt}{\mathcal{F}} \newcommand{\FlexFEA}{\Flex{\bar{0}}} \newcommand{\FlexFEAModified}{\Flex{0}} \newcommand{\FlexOne}{\Flex{1}} \newcommand{\FlexImmersed}{\Flex{\infty}} \newcommand{\FlexInfty}{\mathcal{F}{\infty}} \newcommand{\FlexSpectrum}{\overleftrightarrow{\mathcal{F}}} \newcommand{\FlexSpectrumId}{i} \newcommand{\RegionSymbol}{r} \newcommand{\RegionSet}{\Set{R}} \newcommand{\RegionLocOp}{V} \newcommand{\RegionLocOpTest}{U} \newcommand{\RegionLocOpTrial}{V} \newcommand{\ParallelPartitioningOf}[1]{\From{\Set{P}}{#1}} \newcommand{\ParallelPartitionSet}{\Set{p}} \newcommand{\BasisExtensionTestMat}{\Mat{D}} \newcommand{\BasisExtensionTrialMat}{\Mat{E}} \newcommand{\ActiveIdSet}{\Set{a}} \newcommand{\ConstrainedSet}{\Set{c}} \newcommand{\UnconstrainedTrialSet}{\Set{u}} \newcommand{\UnconstrainedTestSet}{\Set{t}} \newcommand{\QuadraturePointSum}[1]{\sum_{\SymbolQuadraturePoint_{#1}\in\QuadratureSetOf{\Support{\BasisFuncTestSubscripted{\BasisFuncTestId_{#1}}}}}} \newcommand{\TimeStep}{\Delta \SymbolTime} \newcommand{\TimeStepId}{n} \newcommand{\SpectralRadius}{\rho_{\infty}} \newcommand{\SpectralRadiusExplicit}{\rho_{b}} \newcommand{\AlphaF}{\alpha_f} \newcommand{\AlphaM}{\alpha_m} \newcommand{\LineSearchStep}{\alpha} \newcommand{\SymbolTemperature}{\theta} \newcommand{\SymbolThermalExpansionCoefficient}{\alpha} \newcommand{\SymbolThermalStretchRatio}{\vartheta}$$\newcommand{\SymbolContact}{\operatorname{cont}} \newcommand{\VertexAngle}{\theta^\Delta} \newcommand{\stox}{\SurfaceRef \rightarrow \X} \newcommand{\stoa}{\SurfaceRef \rightarrow \SurfaceRef_\lbi} \newcommand{\atox}{\SurfaceRef_\lbi \rightarrow \X} \newcommand{\atos}{\SurfaceRef_\lbi \rightarrow \SurfaceRef} \newcommand{\xtoy}{\X \rightarrow \Y} \newcommand{\xtoa}{\X \rightarrow \SurfaceRef_\lbi} \newcommand{\ytox}{\Y \rightarrow \X} \newcommand{\GenericDeformMapFromContactTessellation}{\ParentOfChild{{\Tessellation}}{\GenericDeformMap}} \newcommand{\VolumeDeformMapFromContactTessellation}{\ManifoldMap{\TessellationOfSurfaceRef}{\VolumeCurr}} \newcommand{\DistYToX}{D} \newcommand{\StrainContact}{R} \newcommand{\StrainContactVariation}{\delta\StrainContact} \newcommand{\PotentialContact}{P^{\epsilon}} \newcommand{\StressContact}{T^{\epsilon}} \newcommand{\StressContactVec}{\Vec{T}^{\epsilon}} \newcommand{\EnergyContact}{\Energy^{\SymbolContact}} \newcommand{\EnergyContactVariation}{\delta\EnergyContact} \newcommand{\ResidualVecFromContactTessellation}{\ParentOfChild{\Tessellation}{\ResidualVec}} \newcommand{\TractionContribution}{\bar{\Vec{f}}} \newcommand{\NormalTractionContribution}{\TractionContribution^{nor}} \newcommand{\TangentialTractionContribution}{\TractionContribution^{tan}} \newcommand{\TangentialRelativeVelocity}{\Vec{v}^{tan}} \newcommand{\FrictionRegularization}{R^{fric}} \newcommand{\UnitTangent}{\Vec{t}} \newcommand{\FrictionCoeff}{c^{fric}} \newcommand{\FrictionRegularizationVelocity}{\epsilon^{fric}}$$\newcommand{\MidGeometryModifier}[1]{\bar{#1}} \newcommand{\SymbolRotationGlobal}{\omega} \newcommand{\SkewMat}[1]{\Mat{s} \left( #1 \right)} \newcommand{\SkewMatInv}[1]{\Mat{s}^{-1} \left( #1 \right)} \newcommand{\Ident}{\Mat{I}} \newcommand{\Cartesian}{\Vec{e}} \newcommand{\ManifoldDim}{n} \newcommand{\LaminaSymbol}{l} \newcommand{\LaminaSymbolUpper}{L} \newcommand{\LaminaMetricSymbol}{j} \newcommand{\ShellMomentSymbol}{q} \newcommand{\DirectorBilinearFunc}[3]{\left(#2,#3\right)_{#1}} \newcommand{\RotObjectiveFunc}{s} \newcommand{\ParamConfig}{\Omega_\xi} \newcommand{\ParamConfigMid}{\Omega_\xi^m} \newcommand{\ParamConfigSec}{\Omega_\xi^s} \newcommand{\ParamConfigBound}{\Gamma_\xi} \newcommand{\ParamConfigMidBound}{\Gamma_\xi^m} \newcommand{\ParamConfigSecBound}{\Gamma_\xi^s} \newcommand{\ParamConfigMuBound}{\Gamma_\xi^\mu} \newcommand{\ParamConfigSigBound}{\Gamma_\xi^\sigma} \newcommand{\ParamConfigDirichMidBound}{\Gamma_\xi^g} \newcommand{\ParamConfigNeuMidBound}{\Gamma_\xi^h} \newcommand{\ParamConfigDirichBound}{\Gamma_\xi^\gamma} \newcommand{\ParamConfigNeuBound}{\Gamma_\xi^\eta} \newcommand{\ParaParam}[1]{\xi_{#1}} \newcommand{\ParaParamVec}{\boldsymbol{\xi}} \newcommand{\RefConfig}{\Omega_0} \newcommand{\RefConfigMid}{\Omega_0^m} \newcommand{\RefConfigSec}{\Omega_0^s} \newcommand{\SurfaceRefRot}{\SurfaceRef^{\SymbolRotationGlobal}} \newcommand{\RefConfigBound}{\Gamma_0} \newcommand{\RefConfigMidBound}{\Gamma_0^m} \newcommand{\RefConfigSecBound}{\Gamma_0^s} \newcommand{\RefConfigMuBound}{\Gamma_0^\mu} \newcommand{\RefConfigSigBound}{\Gamma_0^\sigma} \newcommand{\RefConfigDirichMidBound}{\Gamma_0^g} \newcommand{\RefConfigNeuMidBound}{\Gamma_0^h} \newcommand{\RefConfigDirichBound}{\Gamma_0^\gamma} \newcommand{\RefConfigNeuBound}{\Gamma_0^\eta} \newcommand{\RefMap}{\Vec{X}} \newcommand{\RefMid}{\MidGeometryModifier{\Vec{X}}} \newcommand{\RefDir}{\Vec{D}} \newcommand{\RefSectionPos}{\Vec{R}} \newcommand{\RefCovar}[1]{\Vec{G}_{#1}} \newcommand{\RefMidCovar}[1]{\MidGeometryModifier{\Vec{G}}_{#1}} \newcommand{\RefContra}[1]{\Vec{G}^{#1}} \newcommand{\RefMidContra}[1]{\MidGeometryModifier{\Vec{G}}^{#1}} \newcommand{\RefLamina}[1]{\Vec{L}_{#1}} \newcommand{\RefLaminaDual}[1]{\Vec{\LaminaSymbolUpper}^{#1}} \newcommand{\RefShellProjectedBasis}[1]{\Vec{P}_{#1}} \newcommand{\CurConfig}{\Omega} \newcommand{\CurConfigMid}{\Omega^m} \newcommand{\CurConfigSec}{\Omega^s} \newcommand{\CurConfigBound}{\Gamma} \newcommand{\CurConfigMidBound}{\Gamma^m} \newcommand{\CurConfigSecBound}{\Gamma^s} \newcommand{\CurConfigMuBound}{\Gamma^\mu} \newcommand{\CurConfigSigBound}{\Gamma^\sigma} \newcommand{\CurConfigDirichMidBound}{\Gamma^g} \newcommand{\CurConfigNeuMidBound}{\Gamma^h} \newcommand{\CurConfigDirichBound}{\Gamma^\gamma} \newcommand{\CurConfigNeuBound}{\Gamma^\eta} \newcommand{\CurMap}{\Vec{x}} \newcommand{\CurMid}{\MidGeometryModifier{\Vec{x}}} \newcommand{\CurDir}{\Vec{d}} \newcommand{\CurSectionPos}{\Vec{r}} \newcommand{\CurCovar}[1]{\Vec{g}_{#1}} \newcommand{\CurMidCovar}[1]{\MidGeometryModifier{\Vec{g}}_{#1}} \newcommand{\CurContra}[1]{\Vec{g}^{#1}} \newcommand{\CurMidContra}[1]{\MidGeometryModifier{\Vec{g}}^{#1}} \newcommand{\CurLamina}[1]{\Vec{\LaminaSymbol}_{#1}} \newcommand{\CurLaminaDual}[1]{\Vec{\LaminaSymbol}^{#1}} \newcommand{\CurModLamina}[1]{\hat{\Vec{\LaminaSymbol}}_{#1}} \newcommand{\CurModLaminaDual}[1]{\hat{\Vec{\LaminaSymbol}}^{#1}} \newcommand{\CurShellProjectedBasis}[1]{\Vec{p}_{#1}} \newcommand{\DeformMap}{\varphi} \newcommand{\DeformMapMid}{\varphi^m} \newcommand{\Disp}{\Vec{u}} \newcommand{\DispMid}{\MidGeometryModifier{\Vec{u}}} \newcommand{\RotVec}{\boldsymbol{\omega}} \newcommand{\RotMat}{\boldsymbol{\Lambda}} \newcommand{\RotMatMembrane}{\Mat{\Psi}} \newcommand{\RotMatDelta}{\boldsymbol{\Lambda}_\Delta} \newcommand{\IncRotMat}{\Delta\RotMat} \newcommand{\EulerVec}{\boldsymbol{\vartheta}} \newcommand{\EulerVecNorm}{\theta} \newcommand{\EulerVecSinc}{\boldsymbol{\Theta}} \newcommand{\EulerRotVec}{\boldsymbol{\theta}} \newcommand{\EulerRotAngle}{\theta} \newcommand{\IncEulerRotVec}{\Delta\boldsymbol{\theta}} \newcommand{\AngularVelocity}{\boldsymbol{\eta}} \newcommand{\IncEulerRotVecVel}{\Delta\dot{\boldsymbol{\theta}}} \newcommand{\AngularAcceleration}{\boldsymbol{\alpha}} \newcommand{\IncEulerRotVecAcc}{\Delta\ddot{\boldsymbol{\theta}}} \newcommand{\DeformGrad}{\Mat{F}} \newcommand{\Strain}[1]{\boldsymbol{\tau}_{#1}} \newcommand{\StrainMid}[1]{\boldsymbol{\varepsilon}_{#1}} \newcommand{\StrainCurve}[1]{\boldsymbol{\kappa}_{#1}} \newcommand{\DecompDeform}{\Mat{U}} \newcommand{\ExtraDeformTen}{\Delta\DeformGradTen} \newcommand{\ExtraDeformComp}{\Delta\DeformGradComp} \newcommand{\GreenLagrange}{\Mat{E}} \newcommand{\ShellStrainCurve}[1]{\boldsymbol{\chi}_{#1}} \newcommand{\SecPK}{\Mat{S}} \newcommand{\FirstPK}{\Mat{P}} \newcommand{\Cauchy}{\boldsymbol{\sigma}} \newcommand{\IntTrac}[1]{\Vec{t}_{#1}} \newcommand{\RefModuli}{\Mat{C}} \newcommand{\CurModuli}{\Mat{c}} \newcommand{\PointModuli}{\Mat{m}} \newcommand{\ResModuli}{\Mat{M}} \newcommand{\RefAreaVec}[1]{\Vec{A}_{#1}} \newcommand{\RefAreaMidVec}[1]{\MidGeometryModifier{\Vec{A}}_{#1}} \newcommand{\RefMeas}{G} \newcommand{\UnitNormalRefVec}[1]{\Vec{N}_{#1}} \newcommand{\CurAreaVec}[1]{\Vec{a}_{#1}} \newcommand{\CurAreaMidVec}[1]{\MidGeometryModifier{\Vec{a}}_{#1}} \newcommand{\CurMeas}{g} \newcommand{\UnitNormalCurrVec}[1]{\Vec{n}_{#1}} \newcommand{\CurLaminaMetricTen}{\Mat{\LaminaMetricSymbol}} \newcommand{\CurLaminaMetricComp}{\LaminaMetricSymbol} \newcommand{\CurToModLaminaTransComp}{M} \newcommand{\IntForce}{\Vec{f}^{\, int}} \newcommand{\IntMoment}{\Vec{m}^{int}} \newcommand{\ExtForceBody}{\Vec{f}^{\, ext}_b} \newcommand{\ExtMomentBody}{\Vec{m}^{ext}_b} \newcommand{\ShellExtMomentBody}{\Vec{\ShellMomentSymbol}^{ext}_b} \newcommand{\ExtForceTrac}{\Vec{f}^{\, ext}_h} \newcommand{\ExtMomentTrac}{\Vec{m}^{ext}_h} \newcommand{\ShellExtMomentTrac}{\Vec{\ShellMomentSymbol}^{ext}_h} \newcommand{\MassForce}{\Vec{f}^{\,\rho}} \newcommand{\MassMoment}{\Vec{m}^{\rho}} \newcommand{\ShellMassMoment}{\Vec{\ShellMomentSymbol}^{\rho}} \newcommand{\ResidualInternal}{\Vec{R}^{int}} \newcommand{\ResidualExternal}{\Vec{R}^{ext}} \newcommand{\ResidualMass}{\Mat{R}^{\rho}} \newcommand{\NodalMass}[1]{\Mat{M}_{#1}} \newcommand{\NodalLinearInertia}[1]{A_{#1}} \newcommand{\NodalAngularInertia}[1]{I_{#1}} \newcommand{\ShellDeltaAngle}{\phi} \newcommand{\BStrainMatrix}[2]{\Mat{B}_{#1#2}} \newcommand{\ShellDrillingStiffness}{k} \newcommand{\DrillConstraint}{C} \newcommand{\IntForceDrilling}{\Vec{f}^{d}} \newcommand{\ModuliDrilling}{\Mat{M}^d} \newcommand{\ResidualDrilling}{\Vec{R}^d} \newcommand{\SymbolTop}{top} \newcommand{\SymbolCentroid}{c} \newcommand{\SymbolLeverArm}{r} \newcommand{\SpatialCoordVecCentroid}{\SpatialCoordVec^{\SymbolCentroid}} \newcommand{\SpatialCoordCentroidX}{\SpatialCoordX^{\SymbolCentroid}} \newcommand{\SpatialCoordCentroidY}{\SpatialCoordY^{\SymbolCentroid}} \newcommand{\SpatialCoordCentroidZ}{\SpatialCoordZ^{\SymbolCentroid}} \newcommand{\LeverArm}{\Vec{\SymbolLeverArm}} \newcommand{\LeverArmX}{\SymbolLeverArm_{\SpatialCoordX}} \newcommand{\LeverArmY}{\SymbolLeverArm_{\SpatialCoordY}} \newcommand{\LeverArmZ}{\SymbolLeverArm_{\SpatialCoordZ}} \newcommand{\MomentFirstMass}{Q} \newcommand{\MomentFirstMassDetailed}[1]{Q_{#1}} \newcommand{\MomentSecondMass}{I} \newcommand{\MomentSecondMassDetailed}[1]{I_{#1}} \newcommand{\DistributedLoad}{q} \newcommand{\TracForceCurrCompX}{\TracForceCurrComp_{\SpatialCoordX}} \newcommand{\TracForceCurrCompY}{\TracForceCurrComp_{\SpatialCoordY}} \newcommand{\TracForceCurrCompZ}{\TracForceCurrComp_{\SpatialCoordZ}} \newcommand{\TracMomentCurrCompDetailed}[1]{\TracMomentCurrComp_{#1}} \newcommand{\TracForceCurrCompSupportReaction}{r^{\TracForceCurrComp}} \newcommand{\TracForceCurrCompSupportReactionDetailed}[1]{\TracForceCurrCompSupportReaction_{#1}} \newcommand{\TracForceCurrCompDistributedLoad}{\DistributedLoad^{\TracForceCurrComp}} \newcommand{\TracForceCurrCompPointLoad}{p^{\TracForceCurrComp}} \newcommand{\TracMomentCurrVec}{\Vec{\TracMomentCurrComp}} \newcommand{\TracMomentCurrComp}{m} \newcommand{\BodyForceCurrCompX}{\BodyForceCurrComp_{\SpatialCoordX}} \newcommand{\BodyForceCurrCompY}{\BodyForceCurrComp_{\SpatialCoordY}} \newcommand{\BodyForceCurrCompZ}{\BodyForceCurrComp_{\SpatialCoordZ}} \newcommand{\BodyForceCurrCompDistributedLoad}{\DistributedLoad^{\BodyForceCurrComp}} \newcommand{\BodyForceCurrCompAxialLoad}{f^{\BodyForceCurrComp}} \newcommand{\StressCauchyTractionVec}{\Vec{t}} \newcommand{\StressCauchyTractionX}{t_{\SpatialCoordX}} \newcommand{\StressCauchyTractionY}{t_{\SpatialCoordY}} \newcommand{\StressCauchyTractionZ}{t_{\SpatialCoordZ}} \newcommand{\StressCauchyNormal}{\sigma} \newcommand{\StressCauchyNormalDetailed}[1]{\StressCauchyNormal_{#1}} \newcommand{\StressCauchyNormalXX}{\StressCauchyComp_{\SpatialCoordX\SpatialCoordX}} \newcommand{\StressCauchyNormalYY}{\StressCauchyComp_{\SpatialCoordY\SpatialCoordY}} \newcommand{\StressCauchyNormalZZ}{\StressCauchyComp_{\SpatialCoordZ\SpatialCoordZ}} \newcommand{\StressCauchyShear}{\tau} \newcommand{\StressCauchyShearDetailed}[1]{\StressCauchyShear_{#1}} \newcommand{\StressCauchyShearXY}{\StressCauchyShear_{\SpatialCoordX\SpatialCoordY}} \newcommand{\StressCauchyShearYX}{\StressCauchyShear_{\SpatialCoordY\SpatialCoordX}} \newcommand{\StressCauchyShearXZ}{\StressCauchyShear_{\SpatialCoordX\SpatialCoordZ}} \newcommand{\StressCauchyShearZX}{\StressCauchyShear_{\SpatialCoordZ\SpatialCoordX}} \newcommand{\StressCauchyShearYZ}{\StressCauchyShear_{\SpatialCoordY\SpatialCoordZ}} \newcommand{\StressCauchyShearZY}{\StressCauchyShear_{\SpatialCoordZ\SpatialCoordY}} \newcommand{\StressCauchyResultantForce}{F} \newcommand{\StressCauchyResultantForceDetailed}[1]{F_{#1}} \newcommand{\StressCauchyResultantForceVec}{\Vec{\StressCauchyResultantForce}} \newcommand{\StressCauchyResultantMoment}{M} \newcommand{\StressCauchyResultantMomentDetailed}[1]{M_{#1}} \newcommand{\StressCauchyResultantMomentVec}{\Vec{\StressCauchyResultantMoment}} \newcommand{\StressCauchyResultantForceShear}{V} \newcommand{\StressCauchyResultantForceShearDetailed}[1]{V_{#1}} \newcommand{\StressCauchyResultantForceAxial}{\StressCauchyResultantForce} \newcommand{\StressCauchyPrincipalMax}{\StressCauchyComp_{\mathrm{max}}} \newcommand{\StressCauchyPrincipalMid}{\StressCauchyComp_{\mathrm{mid}}} \newcommand{\StressCauchyPrincipalMin}{\StressCauchyComp_{\mathrm{min}}} \newcommand{\StressCauchyNominal}{\StressCauchyComp_{\mathrm{nom}}} \newcommand{\DispFieldCurrCompX}{\DispFieldCurrComp_{\SpatialCoordX}} \newcommand{\DispFieldCurrCompY}{\DispFieldCurrComp_{\SpatialCoordY}} \newcommand{\DispFieldCurrCompZ}{\DispFieldCurrComp_{\SpatialCoordZ}} \newcommand{\DispFieldMidCurrCompX}{\bar{\DispFieldCurrComp}_{\SpatialCoordX}} \newcommand{\DispFieldMidCurrCompY}{\bar{\DispFieldCurrComp}_{\SpatialCoordY}} \newcommand{\StrainSymGradCompXX}{\StrainSymGradComp_{\SpatialCoordX\SpatialCoordX}} \newcommand{\StrainSymGradCompYY}{\StrainSymGradComp_{\SpatialCoordY\SpatialCoordY}} \newcommand{\StrainSymGradCompZZ}{\StrainSymGradComp_{\SpatialCoordZ\SpatialCoordZ}} \newcommand{\StrainShearComp}{\gamma} \newcommand{\StrainShearCompXY}{\StrainShearComp_{\SpatialCoordX\SpatialCoordY}} \newcommand{\StrainShearCompXZ}{\StrainShearComp_{\SpatialCoordX\SpatialCoordZ}} \newcommand{\StrainShearCompYZ}{\StrainShearComp_{\SpatialCoordY\SpatialCoordZ}} \newcommand{\EquationBeamShearStress}{\frac{\StressCauchyResultantForceShear \MomentFirstMass}{\MomentSecondMassDetailed{\SpatialCoordZ}\Width}} \newcommand{\EquationVecInCoords}{\Vec{v} = v_{\SpatialCoordX} \BasisVecI + v_{\SpatialCoordY} \BasisVecJ + v_{\SpatialCoordZ} \BasisVecK} \newcommand{\EquationMatVecMultiply}{\Mat{M}\Vec{v} = \begin{bmatrix} \DotProduct{\Vec{m}_1}{\Vec{v}} \\ \DotProduct{\Vec{m}_2}{\Vec{v}} \\ \DotProduct{\Vec{m}_3}{\Vec{v}} \end{bmatrix}} \newcommand{\EquationMatMatMultiply}{\Mat{M}\Mat{N} = \begin{bmatrix} \DotProduct{\Vec{m}_1}{\Mat{n}_1} & \DotProduct{\Vec{m}_1}{\Mat{n}_2} & \DotProduct{\Vec{m}_1}{\Mat{n}_3} \\ \DotProduct{\Vec{m}_2}{\Mat{n}_1} & \DotProduct{\Vec{m}_2}{\Mat{n}_2} & \DotProduct{\Vec{m}_2}{\Mat{n}_3} \\ \DotProduct{\Vec{m}_3}{\Mat{n}_1} & \DotProduct{\Vec{m}_3}{\Mat{n}_2} & \DotProduct{\Vec{m}_3}{\Mat{n}_3} \end{bmatrix}} \newcommand{\Force}{F} \newcommand{\ShearFlow}{f}$$\newcommand{\BaseSymbol}{b} \newcommand{\EmbeddedSymbol}{\Theta} \newcommand{\EmbeddedAtlas}{\From{\EmbeddedSymbol}{\ParametricAtlas}} \newcommand{\EmbeddedPhysicalAtlas}{\ManifoldFrom{\EmbeddedSymbol}} \newcommand{\RefinedSymbol}{r} \newcommand{\HierarchicalSymbol}{\textsf{H}} \newcommand{\BlockSymbol}{\textsf{B}} \newcommand{\BaseModifier}[1]{{#1}_{\BaseSymbol}} \newcommand{\EmbeddedModifier}[1]{{#1}_{\EmbeddedSymbol}} \newcommand{\RefinedModifier}[1]{{#1}_{\RefinedSymbol}} \newcommand{\HierarchicalModifier}[1]{{#1}_{\HierarchicalSymbol}} \newcommand{\BlockModifier}[1]{{#1}_{\BlockSymbol}} \newcommand{\RefinementLevelI}{i} \newcommand{\RefinementLevelJ}{j} \newcommand{\RefinementLevelK}{k} \newcommand{\HighestRefinementLevel}{N} \newcommand{\BaseParentDomain}{\BaseModifier{\ParentDomain}} \newcommand{\RefinedDomain}[1]{\ParamDomain_{#1}} \newcommand{\BaseMesh}{\BaseModifier{\Topology}} \newcommand{\BaseBezierMesh}{\BaseModifier{\Mesh}} \newcommand{\EmbeddedMesh}{\EmbeddedModifier{\Topology}} \newcommand{\RefinedMesh}{\RefinedModifier{\Topology}} \newcommand{\RefinedBezierMesh}{\RefinedModifier{\Mesh}} \newcommand{\HierarchicalBezierMesh}{\HierarchicalModifier{\Mesh}} \newcommand{\HierarchicalUsplineMesh}{\HierarchicalModifier{\MeshAdmissible}} \newcommand{\RefinementLevelMesh}[1]{\Mesh_{#1}} \newcommand{\LeafElemSet}[1]{\Set{L}_{#1}} \newcommand{\ActiveElemSet}[1]{\Set{A}_{#1}} \newcommand{\ActiveBasisFuncs}[1]{\Set{UFA}_{#1}} \newcommand{\UnderRefinedBasisFuncs}[1]{\Set{UFU}_{#1}} \newcommand{\HierarchicalBasisFuncs}{\Set{UFH}} \newcommand{\TruncatedBasisFuncs}{\Set{UFT}} \newcommand{\FuncsTruncatedToLevel}[1]{\Set{TK}_{#1}} \newcommand{\FuncsTouchingActiveFuncs}[1]{\Set{T}_{#1}} \newcommand{\FuncsSubsetOfCoarserFuncs}[1]{\Set{S}_{#1}} \newcommand{\RefinementLevelBasis}[1]{\GlobalBasisFuncSetOnQuantity{\RefinementLevelMesh{#1}}} \newcommand{\RefinementLevelSpace}[1]{\From{\SpaceGlobal}{\RefinementLevelMesh{#1}}} \newcommand{\BaseAssociated}{\operatorname{\textsf{base}}} \newcommand{\BaseAssociatedOf}[1]{\Of{\BaseAssociated}{#1}} \newcommand{\TruncationOp}[2]{\Of{\operatorname{\textsf{trunc}}_{#1}}{#2}} \newcommand{\EmbeddedToBaseChart}{\EmbeddedModifier{\ParentDomainChart}} \newcommand{\InvEmbeddedToBaseChart}{(\EmbeddedToBaseChart)^{-1}} \newcommand{\RefinementCoefficients}{c^A_B} \newcommand{\RefinementMat}{\Mat{D}} \newcommand{\NthLevelProlongationMat}[1]{\ProlongationMat_{#1}} \newcommand{\NthLevelActiveMat}[1]{\Mat{A}_{#1}} \newcommand{\NthLevelRefinementMat}[1]{\RefinementMat_{#1}} \newcommand{\NthLevelTruncationMat}[1]{\Mat{T}_{#1}} \newcommand{\NthLevelBasisMat}[1]{\Mat{S}_{#1}} \newcommand{\NthLevelMassMat}[1]{\MassMat_{#1}} \newcommand{\FuncIndex}{\operatorname*{\mathsf{ID}}} \newcommand{\MultiLevelRefinementMat}[2]{\RefinementMat_{#1 #2}} \newcommand{\BlockRefinementMat}{\BlockModifier{\RefinementMat}} \newcommand{\BlockMassMat}{\BlockModifier{\MassMat}}$$\newcommand{\Topology}{\mathsf{T}} \newcommand{\UTopology}{\mathsf{U}} \newcommand{\TopoAlpha}[1]{\alpha_{#1}} \newcommand{\TopoPhi}[1]{\phi_{#1}} \newcommand{\Map}[1]{\mathsf{#1}} \newcommand{\OMap}[1]{\mathsf{#1}} \newcommand{\UOMap}[1]{\UTopology(#1)} \newcommand{\DartsOf}[1]{\operatorname*{\mathsf{D}}\left(#1\right)} \newcommand{\DartIndex}[1]{\operatorname*{\mathsf{ID}}\left(#1\right)} \newcommand{\Decomp}{\operatorname{dec}} \newcommand{\DistFunc}[2]{d(#1,#2)} \newcommand{\Inside}{\Omega} \newcommand{\ImplicitFuncSymbol}{f} \newcommand{\ImplicitFunc}[1]{\ImplicitFuncSymbol(#1)} \newcommand{\ParametricCurve}{\bar{\Curve}} \newcommand{\ParametricSurface}{\bar{\Surface}} \newcommand{\SamplePoint}[1]{t_{#1}} \newcommand{\SurfaceParamDomain}{\From{\ParamDomain}{\Surface}} \newcommand{\ExteriorDeriv}{\mathop{\mathbf{d}}} \newcommand{\RejectOp}[1]{\mathop{\textrm{Rej}_{#1}}} \newcommand{\Reject}[2]{\RejectOp{#1}\left(#2\right)} \newcommand{\ReflectOp}[1]{\mathop{\textrm{Ref}_{#1}}} \newcommand{\Reflect}[2]{\ReflectOp{#1}\left(#2\right)} \newcommand{\SignOp}{\mathop{\textrm{sign}}} \newcommand{\Sign}[1]{\SignOp\left(#1\right)} \newcommand{\TangentOne}{\Vec{t}_1} \newcommand{\TangentTwo}{\Vec{t}_2} \newcommand{\TangentThree}{\Vec{t}_3} \newcommand{\KForm}[1]{\Vec{#1}} \newcommand{\TangentDualOne}{\KForm{\tau}_1} \newcommand{\TangentDualTwo}{\KForm{\tau}_2} \newcommand{\TangentDualThree}{\KForm{\tau}_3} \newcommand{\SubmeshIndex}{i} \newcommand{\NumberOfSubmeshes}{n^{\StructuredSymbol}} \newcommand{\NonzeroSet}[1]{\mathsf{N}_{#1}} \newcommand{\OverlapSet}[1]{\mathsf{O}_{#1}} \newcommand{\TensorProductIndex}{i} \newcommand{\TensorProductIndexAlt}{j} \newcommand{\SelectedIndexSet}{\Set{S}} \newcommand{\NotSelectedIndexSet}{\bar{\Set{S}}} \newcommand{\SubDomSupport}{\operatorname{\mathsf{sds}}} \newcommand{\TensorProductBasisFunc}{\GlobalBasisFuncDetailed{}{\GlobalBasisFuncIdVec}} \newcommand{\ComponentBasisFunc}[1]{\GlobalBasisFuncDetailed{#1}{\GlobalBasisFuncId_{#1}}} \newcommand{\SelectedIndicesBasisFunc}{\GlobalBasisFuncDetailed{\SelectedIndexSet}{\GlobalBasisFuncIdVec_{\SelectedIndexSet}}} \newcommand{\GlobalBasisFuncIdOne}{\GlobalBasisFuncId_0} \newcommand{\GlobalBasisFuncIdTwo}{\GlobalBasisFuncId_1} \newcommand{\GlobalBasisFuncIdVec}{\Vec{A}}$