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type, title, complexity, domain, aliases, created, updated, address, tags, status, related, sources
| type | title | complexity | domain | aliases | created | updated | address | tags | status | related | sources | ||||||||||||
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| concept | Solid Element Stiffness Integration | advanced | computational-mechanics |
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2026-05-28 | 2026-05-28 | c-000052 |
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current |
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Solid Element Stiffness Integration
Definition
Solid element stiffness integration evaluates the element stiffness matrix for a three-dimensional continuum element by numerically integrating B^T D B over the element volume.
How It Works
The source uses the standard displacement-based stiffness form:
K = integral_V B^T D B dV
= integral B^T D B |J| dxi deta dzeta
Here B is the Solid Element Strain-Displacement Matrix, D is the three-dimensional Hooke-law constitutive matrix, and |J| is the determinant of the Jacobian that maps the natural-coordinate integration region to physical volume.
The notes list quadrature schemes for the first-order solid topologies: one-point integration for the 4-node tetrahedron, eight-point integration for the 5-node pyramid, six-point integration for the 6-node wedge, and eight-point integration for the 8-node hexahedron.
Why It Matters
The stiffness integral is where interpolation, material law, element distortion, and numerical quadrature meet. Incorrect quadrature or a poor Jacobian can produce inaccurate stiffness, spurious mechanisms, or poor convergence even when the symbolic formulation is correct.
Connections
- Isoparametric Finite Elements supplies the natural-coordinate integration framework.
- Solid Element Shape Functions and Solid Element Strain-Displacement Matrix define the integrand.
- Incompatible Mode Solid Elements modifies the displacement field and therefore expands the stiffness matrix before static condensation.