13.13. PLANE13 - 2-D Coupled-Field Solid

Matrix or VectorGeometryShape FunctionsIntegration Points
Magnetic Potential Coefficient Matrix; and Permanent Magnet and Applied Current Load VectorQuadEquation 11–1232 x 2
TriangleEquation 11–103

1 if planar
3 if axisymmetric

Thermal Conductivity Matrix QuadEquation 11–128Same as coefficient matrix
TriangleEquation 11–108
Stiffness Matrix; and Thermal and Magnetic Force Load VectorQuadEquation 11–120 and Equation 11–121 and, if modified extra shapes are included (KEYOPT(2) = 0) and element has 4 unique nodes) Equation 11–132 and Equation 11–133.Same as coefficient matrix
TriangleEquation 11–100 and Equation 11–101
Mass and Stress Stiffness MatricesQuadEquation 11–120 and Equation 11–121Same as coefficient matrix
TriangleEquation 11–100 and Equation 11–101
Specific Heat MatrixSame as conductivity matrix. Matrix is diagonalized as described in Lumped MatricesSame as coefficient matrix
Damping (Eddy Current) MatrixQuadEquation 11–123 and Equation 11–129Same as coefficient matrix
TriangleEquation 11–103 and Equation 11–109
Convection Surface Matrix and Load VectorSame as conductivity matrix, specialized to the surface2
Pressure Load VectorSame as mass matrix specialized to the face2
Load TypeDistribution
Current DensityBilinear across element
Current Phase AngleBilinear across element
Heat GenerationBilinear across element
PressureLinear along each face

References: Wilson([38]), Taylor, et al.([49]), Silvester, et al.([72]),Weiss, et al.([94]), Garg, et al.([95])

13.13.1. Other Applicable Sections

Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations. Heat Flow describes the derivation of thermal element matrices and load vectors as well as heat flux evaluations. Derivation of Electromagnetic Matrices and Electromagnetic Field Evaluations discuss the magnetic vector potential method, which is used by this element. The diagonalization of the specific heat matrix is described in Lumped Matrices.

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