Coreform Product Documentation

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\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{\BoundingVolumeTreeNode}{N} \newcommand{\AxisAlignedBoundingBox}[1]{\Of{\Set{AABB}}{#1}} \newcommand{\AxisAlignedBoundingBoxLower}[1]{c^l_{#1}} \newcommand{\AxisAlignedBoundingBoxLowerVec}{\Vec{c}^l} \newcommand{\AxisAlignedBoundingBoxUpper}[1]{c^u_{#1}} \newcommand{\AxisAlignedBoundingBoxUpperVec}{\Vec{c}^u} \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{\PartitionUnityConstraintLM}{\Lambda} \newcommand{\ParametricConstraintLMComp}{\LagrangeMultCurrComp} \newcommand{\ParametricConstraintLMTen}{\LagrangeMultCurrTen} \newcommand{\ParametricConstraintSlack}{\sigma} \newcommand{\ParametricConstraintSlackTen}{\Vec{\sigma}} \newcommand{\InverseMapTangentComp}{T} \newcommand{\InverseMapTangent}{\Mat{T}} \newcommand{\LowerBound}[2]{B^l\left(#1,#2\right)} \newcommand{\UpperBound}[2]{B^u\left(#1,#2\right)} \newcommand{\UpperBoundValue}{U} \newcommand{\SimplexTangentSpaceBasis}[1]{\Vec{B}_{#1}} \newcommand{\SimplexTangentSpaceBasisTrad}[1]{\SimplexTangentSpaceBasis{#1}^t} \newcommand{\SimplexTangentSpaceBasisOrth}[1]{\SimplexTangentSpaceBasis{#1}^o} \newcommand{\SimplexTangentSpaceBasisPerm}[1]{\SimplexTangentSpaceBasis{#1}^p} \newcommand{\SimplexTangentSpaceBasisLagrange}[1]{\SimplexTangentSpaceBasis{#1}^{\lambda}} \newcommand{\SimplexBasisTransformPerm}[2]{\Mat{C}_{#1#2}} \newcommand{\ParamCoordTradComp}[1]{\ParamCoordComp^{t}_{#1}} \newcommand{\ParamCoordPermComp}[1]{\ParamCoordComp^{p}_{#1}} \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{\SymbolDispPrescribed}{g} \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{\SymbolPenaltyDimensionless}{c_{\SymbolPenalty}} \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{\DispPrescribedFieldCurrComp}{\SymbolDispPrescribed} \newcommand{\DispFieldCurrTen}{\Vec{\DispFieldCurrComp}} \newcommand{\DispPrescribedFieldCurrTen}{\Vec{\DispPrescribedFieldCurrComp}} \newcommand{\VelFieldCurrTen}{\dot{\DispFieldCurrTen}} \newcommand{\VelPrescribedFieldCurrTen}{\dot{\DispPrescribedFieldCurrTen}} \newcommand{\AccFieldCurrTen}{\ddot{\DispFieldCurrTen}} \newcommand{\AccFieldCurrComp}{\ddot{\DispFieldCurrComp}} \newcommand{\AccPrescribedFieldCurrTen}{\ddot{\DispPrescribedFieldCurrTen}} \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{\DeformCauchyGreenLeftElasticTen}{\DeformCauchyGreenLeftTen^\KinematicElast} \newcommand{\DeformCauchyGreenLeftElasticTenWithoutI}{\DeformCauchyGreenLeftTen^{\KinematicElast\Delta}} \newcommand{\DeformCauchyGreenLeftElasticTenDef}{\DeformCauchyGreenLeftElasticTen \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{\MaterialYoungsModulusEstimate}{E_{\text{est}}} \newcommand{\MaterialPoissonsRatio}{\nu} \newcommand{\MaterialMassDensity}{\rho} \newcommand{\MaterialBulkModulus}{\kappa} \newcommand{\MaterialBulkModulusEstimate}{\MaterialBulkModulus_{\text{est}}} \newcommand{\MaterialBulkModulusTangent}{\tilde{\MaterialBulkModulus}} \newcommand{\MaterialShearModulus}{\mu} \newcommand{\MaterialShearModulusEstimate}{\MaterialShearModulus_{\text{est}}} \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{\MaxTimeStep}{\TimeStep_{\max}} \newcommand{\MinTimeStep}{\TimeStep_{\min}} \newcommand{\TimeStepDecreaseFactor}{c_{\text{dec}}} \newcommand{\TimeStepIncreaseFactor}{c_{\text{inc}}} \newcommand{\CriticalTimeStep}{\TimeStep_{\text{cr}}} \newcommand{\TimeStepSafetyFactor}{s} \newcommand{\MassDampingCoeff}{\alpha} \newcommand{\DampingRatio}{\xi} \newcommand{\AngularFrequency}{\omega} \newcommand{\VibrationalModeVec}{\Vec{\Phi}} \newcommand{\MaxVibrationalModeVec}{\Vec{\Phi}_{\text{max}}} \newcommand{\MaxFrequency}{\AngularFrequency_{\text{max}}} \newcommand{\RayleighQuotient}{R} \newcommand{\BifurcationTimeStep}{\TimeStep_{b}} \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{\StabilizeParam}{\tau} \newcommand{\StabilizeConstant}{c_{\StabilizeParam}}$$\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{\ActivationDist}{\epsilon} \newcommand{\Triangle}[1]{\mathbf{T}_{#1}} \newcommand{\Ball}[2]{\mathbf{B}_{#1}\left(#2\right)} \newcommand{\Intersection}{\cap} \newcommand{\GenericDeformMapFromContactTessellation}{\ParentOfChild{{\Tessellation}}{\GenericDeformMap}} \newcommand{\VolumeDeformMapFromContactTessellation}{\ManifoldMap{\TessellationOfSurfaceRef}{\VolumeCurr}} \newcommand{\DistYToX}{D} \newcommand{\ModifiedDistYToX}{\hat{\DistYToX}} \newcommand{\EffDistYToX}{\DistYToX^{eff}} \newcommand{\SmoothEffDistYToX}{\tilde{\DistYToX}^{eff}} \newcommand{\StrainContact}{R} \newcommand{\StrainContactVariation}{\delta\StrainContact} \newcommand{\ContactForceScale}{\alpha^{FS}} \newcommand{\PotentialContact}{P^{\epsilon}} \newcommand{\StressContact}{T^{\epsilon}} \newcommand{\StressContactVec}{\Vec{T}^{\epsilon}} \newcommand{\ContactStiffness}{\kappa} \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{\DispGrad}{\Mat{G}} \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}}$