Finite Element & Meshing Reference

Solver Element Catalog (IDs 1–13 & CFD)

WebCAE incorporates a rigorous, mathematically verified finite element and finite volume library compiled to WebAssembly for zero-installation browser execution and compiled natively for multi-core desktop workstations. The internal structural solver adheres to the exact integer types defined in enum StructuralElemType in solver_structural.hpp (IDs 1 through 13), alongside cell-centered/dual polyhedral finite volumes for Navier–Stokes CFD.

#1 STRUCT_BEAM — 3D Timoshenko Beam (Basic)

The STRUCT_BEAM element is a two-node 3D beam formulation with 6 engineering degrees of freedom per node ($u_x, u_y, u_z, \theta_x, \theta_y, \theta_z$), yielding 12 DOFs per element.

• Material Vector: Compact 6-parameter vector: [E, G, A, Iy, Iz, J] where $E$ is Young's modulus, $G$ is shear modulus, $A$ is cross-sectional area, $I_y, I_z$ are principal moments of inertia, and $J$ is Saint-Venant torsional constant.
• Transverse Shear Deformation: Based on Timoshenko beam theory. Cross-sectional rotation is kinematically decoupled from the centerline deflection slope. Shear deformation correction coefficients $\Phi_y$ and $\Phi_z$ are computed as: $$\Phi_y = \frac{12 E I_z}{\kappa_y G A L^2}, \quad \Phi_z = \frac{12 E I_y}{\kappa_z G A L^2}$$ where $\kappa_y, \kappa_z$ default to standard shear factors ($5/6$ for rectangular cross-sections). As $L/h \to \infty$, $\Phi \to 0$, naturally recovering Euler–Bernoulli bending without ill-conditioning.
• Torsion: Saint-Venant uniform torsional stiffness $G J / L$ operates about the element local longitudinal axis.
• Corotational Nonlinearity (`beamcorot`): In geometrically nonlinear analyses (nlgeom), the element uses an $SE(3)$ corotational decomposition. Rigid-body translations and finite spatial rotations are filtered into a continuously updating local coordinate triad, while elastic strains are evaluated using linear constitutive relations in the corotated frame.

#2 STRUCT_SHELL_QUAD — 4-Node MITC4 Shell (Basic)

The STRUCT_SHELL_QUAD element is a 4-node quadrilateral shell with 6 DOFs per node (24 DOFs total), implementing Bathe & Dvorkin's Mixed Interpolation of Tensorial Components (MITC4).

• Material Vector: Standard 3-parameter isotropic property set: [E, nu, t] (Young's modulus, Poisson's ratio, sheet thickness).
• Locking Alleviation: In thin shells ($t/L < 0.001$), conventional displacement-based plate elements exhibit severe shear locking. The MITC4 formulation interpolates transverse shear strains $\gamma_{xz}$ and $\gamma_{yz}$ from tying points located at the four midside edges: $$\gamma_{r3} = \frac{1}{2}(1 + s) \gamma_{r3}^{(A)} + \frac{1}{2}(1 - s) \gamma_{r3}^{(C)}$$ $$\gamma_{s3} = \frac{1}{2}(1 + r) \gamma_{s3}^{(B)} + \frac{1}{2}(1 - r) \gamma_{s3}^{(D)}$$ This formulation completely eliminates shear and membrane locking across all slenderness ratios, maintaining consistent accuracy for $L/t > 1000$ (MacNeal–Harder and Scordelis–Lo roof benchmarks).
• Drilling Stabilization: In-plane drilling rotation $\theta_z$ is stabilized via a Hughes–Brezzi penalty parameter $k_{\text{drill}} = 10^{-3} G A_{\text{elem}} t$.

#3 STRUCT_SOLID_HEX — 8-Node Hexahedron (Hex8)

The STRUCT_SOLID_HEX element is an 8-node trilinear isoparametric hexahedral brick with 3 translational DOFs per node ($u_x, u_y, u_z$), totaling 24 DOFs.

• Selective Volumetric Projection ($ar{B}$ / B-bar Method): Standard full $2 \times 2 \times 2$ Gauss integration suffers from severe volumetric locking in nearly incompressible elasticity (Poisson's ratio $\nu \to 0.5$, such as rubber, biological tissue, or unconstrained plastic flow). The WebCAE implementation splits the strain-displacement operator $\mathbf{B}$ into deviatoric and dilatational parts: $$\mathbf{B} = \mathbf{B}_{\text{dev}} + \bar{\mathbf{B}}_{\text{vol}}, \quad \bar{\mathbf{B}}_{\text{vol}} = \frac{1}{V_e} \int_{V_e} \mathbf{B}_{\text{vol}} dV$$ The volumetric strain is projected as constant across the element, eliminating artificial stiffening while preserving full rank ($2 \times 2 \times 2$ deviatoric integration prevents hourglass modes).
• Geometric Nonlinearity (`solidtl`): Evaluated under Total Lagrangian kinematics using the Green–Lagrange strain tensor $\mathbf{E} = \frac{1}{2}(\mathbf{F}^T \mathbf{F} - \mathbf{I})$ and second Piola–Kirchhoff stress $\mathbf{S}$.

#4 STRUCT_SOLID_TET — 4-Node Linear Tetrahedron (Tet4)

The STRUCT_SOLID_TET element is a 4-node linear tetrahedral simplex with 3 translational DOFs per node (12 DOFs total).

• Quadrature: 1 Gauss integration point located at the element centroid ($\xi = \eta = \zeta = 1/4$, weight $W = 1/6$).
• Kinematics: Linear displacement field resulting in a constant strain state across the volume.
• Meshing Role: Serves as the fundamental building block of automatic 3D Delaunay and Advancing Front meshing algorithms (Netgen engine). Well-conditioned for pure tension/compression, thermal conduction, and adaptive $h$-refinement.

#5 STRUCT_MASS — Concentrated Lumped Mass & Rotary Inertia

The STRUCT_MASS element is a 1-node specialized element carrying 6 dynamic degrees of freedom ($u_x, u_y, u_z, \theta_x, \theta_y, \theta_z$).

• Material Vector: 4 values: [m, Ixx, Iyy, Izz] representing point mass $m$ and principal rotary moments of inertia about the coordinate axes.
• Formulation: Contributes strictly to the global mass matrix $\mathbf{M}$ ($m$ to translational DOFs, $I_{xx}, I_{yy}, I_{zz}$ to rotational DOFs). Its stiffness contribution is identically zero ($\mathbf{K} = \mathbf{0}$).
• Application: Used in modal eigenvalue analysis (solve modal) and transient dynamics to model concentrated payloads, engines, actuators, and ballast without requiring complex 3D solid geometries.

#6 STRUCT_SOLID_TET10 — 10-Node Quadratic Curved Tetrahedron (Tet10)

The STRUCT_SOLID_TET10 element is a 10-node quadratic isoparametric tetrahedron with 4 corner vertices and 6 midside edge nodes, each carrying 3 translational DOFs (30 DOFs total).

• Quadrature: Integrated using the 4-point Keast symmetric quadrature rule, integrating quadratic strain energy polynomials exactly.
• Bending & Locking Immunity: Because the displacement field varies quadratically, the strain and stress fields vary linearly across the element. This completely eliminates artificial shear locking under bending loads, achieving benchmark accuracy even with 1–2 elements across thin wall sections.
• Curved CAD Boundary Projection: When meshing CAD models (STEP/IGES/B-Rep), all midside nodes located on CAD curved faces and fillets are projected onto the exact B-Rep surface. This avoids polygonal chord errors and guarantees smooth stress concentration factors ($K_t$) without surface faceting.

#7 STRUCT_SHELL_TRI — 3-Node Mindlin–Reissner Triangular Shell

The STRUCT_SHELL_TRI element is a 3-node flat shell element with 6 DOFs per node (18 DOFs total), coupling membrane, bending, and drilling kinematics:

• Material Vector: Standard 3-parameter property set: [E, nu, t].
• Membrane & Bending: Constant Strain Triangle (CST) formulation combined with Discrete Kirchhoff Triangle (DKT) / Mindlin–Reissner bending constraints.
• Drilling Degree of Freedom ($ heta_z$): Uses Hughes–Brezzi artificial drilling formulation ($k_{\text{drill}} = \gamma G A_{\text{tri}} t$, $\gamma = 10^{-3}$), guaranteeing positive-definiteness for arbitrary 3D spatial folds and smooth curved surfaces.
• Application: Ideal for unstructured triangular meshes generated directly from complex CAD STEP/B-Rep surfaces, organic geometry, and mesh transitions where quad meshing leaves triangular residuals.

#8 STRUCT_SOLID_HEX20 — 20-Node Quadratic Serendipity Hex (Hex20)

The STRUCT_SOLID_HEX20 element is a 20-node quadratic serendipity brick with 8 corner nodes and 12 midside edge nodes (60 DOFs total).

• Quadrature: Integrated via a full $3 \times 3 \times 3 = 27$ Gauss point grid (or reduced 14-point rule).
• Curvature Capture: Midside nodes along curved boundaries snap to CAD cylindrical, toroidal, and spherical curves.
• Accuracy: Serves as the gold standard for notch fatigue evaluation, hole collars, and non-conformal Mortar contact interfaces where stress gradients $\nabla \boldsymbol{\sigma}$ are steep.

#9 STRUCT_SOLID_WEDGE6 — 6-Node Triangular Prism / Wedge (Wedge6)

The STRUCT_SOLID_WEDGE6 element is a 6-node isoparametric prism featuring two parallel 3-node triangular end caps connected by three 4-node quadrilateral lateral faces (18 DOFs total).

• Quadrature: 6 Gauss integration points composed of a 3-point symmetric rule in the triangular base plane multiplied by a 2-point Gauss rule along the prism height.
• Applications: 2.5D Swept meshing of mixed triangular-quadrilateral surfaces, boundary layer extrusion along wall normals (Inflation layers), and transition zones between hexahedral blocks and tetrahedral domains.

#10 STRUCT_SOLID_PYR5 — 5-Node Quadrilateral Pyramid (Pyr5)

The STRUCT_SOLID_PYR5 element is a 5-node pyramid consisting of a 4-node quadrilateral base and a single apex vertex (15 DOFs total).

• Rational Shape Functions: Standard polynomial shape functions degenerate at the apex vertex. STRUCT_SOLID_PYR5 utilizes Bedrosian–Dhondt rational shape functions: $$N_i(\xi, \eta, \zeta) = \frac{1}{4} \left( (1 + \xi_i \xi)(1 + \eta_i \eta) - \zeta + \frac{\xi_i \xi \eta_i \eta \zeta}{1 - \zeta} \right) \quad (i=1..4), \quad N_5 = \zeta$$ ensuring strict $C^0$ continuity, geometric isotropy, and partition of unity throughout the element domain.
• Topological Bridge: Seamlessly connects the 4-node quad face of a Hex8 or Wedge6 to four 3-node triangular faces of adjacent Tet4 elements without hanging nodes or MPCs.

#11 STRUCT_SOLID_HEX27 — 27-Node Complete Lagrange Hexahedron (Hex27)

The STRUCT_SOLID_HEX27 element is a 27-node complete quadratic brick featuring 8 corner nodes, 12 midside edge nodes, 6 face center nodes, and 1 interior volume centroid node (81 DOFs total).

• Complete Tensor-Product Basis: Based on the full triquadratic Lagrange polynomial product $Q_2 = P_2(\xi) \otimes P_2(\eta) \otimes P_2(\zeta)$, eliminating the serendipity family's known deficiencies under severe interior geometric distortion.
• Quadrature: $3 \times 3 \times 3 = 27$ Gauss points.
• Application: Used in high-precision research benchmarks, higher-order acoustic and elastic wave propagation, and p-version finite element validation.

#12 STRUCT_BEAM_EXT — 3D Timoshenko Extended Beam

The STRUCT_BEAM_EXT element is an extended 2-node 3D Timoshenko beam (12 DOFs) governed by a 22-value material and kinematic array in elemMat:

• Layout of `elemMat[22]`: - [0..5]: Core cross-section properties: E, G, A, Iy, Iz, J - [6..7]: Exact shear correction factors $k_{sy}, k_{sz}$ - [8]: Product moment of area $I_{yz}$ for asymmetric cross-sections (L-angles, Z-purlins) - [9]: Section roll angle $\beta$ in radians (rollDeg) - [10..12]: Custom reference orientation vector $\mathbf{v}_{\text{ref}}$ for the local $z$-axis - [13..15]: Rigid eccentric arm offset at node $I$: $\mathbf{e}_1 = [e_{x1}, e_{y1}, e_{z1}]$ - [16..18]: Rigid eccentric arm offset at node $J$: $\mathbf{e}_2 = [e_{x2}, e_{y2}, e_{z2}]$ - [19..20]: Transverse shear center offset relative to centroid: $(y_s, z_s)$ - [21]: Non-structural distributed line mass $NSM$ (kg/m)
• Steiner Parallel-Axis Coupling: When attached to shell surfaces (assign beam, flush=outer|inner), the kinematic arm transformation matrix $\mathbf{A}$ automatically couples beam axial forces to shell out-of-plane bending moments, producing the exact effective inertia $I_{\text{eff}} = I_s + I_b + A_{\text{eff}} e^2$.

#13 STRUCT_SHELL_QUAD_L — 4-Node MITC4 Composite Laminate Shell

The STRUCT_SHELL_QUAD_L element is an extended 4-node MITC4 shell (24 DOFs) designed for layered composite laminates, orthotropic materials, and offset reference surfaces, governed by 36 values in elemMat:

• Layout of `elemMat[36]`: - [0..5]: Membrane stiffness matrix $\mathbf{A}$ ($A_{11}, A_{12}, A_{13}, A_{22}, A_{23}, A_{33}$) - [6..11]: Membrane-bending coupling matrix $\mathbf{B}$ ($B_{11}, B_{12}, B_{13}, B_{22}, B_{23}, B_{33}$) - [12..17]: Bending stiffness matrix $\mathbf{D}$ ($D_{11}, D_{12}, D_{13}, D_{22}, D_{23}, D_{33}$) - [18..20]: Transverse shear stiffness matrix $\mathbf{A}_s$ ($A_{s11}, A_{s12}, A_{s22}$) with shear factors - [21]: Total physical laminate thickness $h$ (m) - [22..24]: Unit material principal axis vector $\mathbf{x}_e$ in global coordinates - [25..27]: Mass moments per unit area: $I_0 = \int \rho dz$, $I_1 = \int \rho z dz$, $I_2 = \int \rho z^2 dz$ - [28]: Drilling stiffness scaling factor - [29]: Non-structural surface mass $NSM$ (kg/m²) - [30..32]: Thermal membrane force vector $\mathbf{N}_T$ - [33..35]: Thermal bending moment vector $\mathbf{M}_T$
• Midsurface Offset $z_0$: Offsetting the reference surface by $z_0$ automatically updates the coupling matrices without changing mesh geometry: $\mathbf{B}' = \mathbf{B} + z_0 \mathbf{A}$ and $\mathbf{D}' = \mathbf{D} + 2 z_0 \mathbf{B} + z_0^2 \mathbf{A}$.

CFD FVM (SU2_CFD) — Finite Volume Dual Polyhedral Formulation

In computational fluid dynamics mode (cfd), WebCAE executes an embedded native build of SU2_CFD using a vertex-centered median dual finite volume method:

• Dual Control Volumes: A dual polyhedral control volume $\Omega_i$ is constructed around each mesh node by joining the centroids of surrounding elements (tetrahedra, prisms, pyramids, or hexahedra) to the midpoints of surrounding edges and faces.
• Governing Equations: Conservative Navier–Stokes conservation laws: $$\frac{\partial}{\partial t} \int_{\Omega_i} \mathbf{U} d\Omega + \sum_{j \in N(i)} \oint_{\partial \Omega_{ij}} (\mathbf{F}_c - \mathbf{F}_v) \cdot d\mathbf{A} = \int_{\Omega_i} \mathbf{Q} d\Omega$$
• Convective Fluxes: Roe approximate Riemann solver and JST (Jameson–Schmidt–Turkel) centered schemes with adaptive 2nd/4th-order artificial dissipation.
• Turbulence Models: - Menter SST ($k-\omega$): Two-equation model blending $k-\omega$ near walls with $k-\epsilon$ in the far field, capturing adverse pressure gradients and flow separation. - Spalart–Allmaras: Robust one-equation model optimized for external aerospace aerodynamics.
• Dynamic Meshes & Turbomachinery: - Sliding Mesh: Non-conformal conservative interpolation across rotating rotor/stator interfaces. - ALE (Arbitrary Lagrangian–Eulerian): Elastic mesh morphing for deforming boundaries and 1-DOF fluid-structure interaction (flutter, rotor spin-up).

Automatic Element Selection (Why There Is No Raw ID Selector in UI)

In WebCAE, users never need to memorize internal integer IDs (1 through 13) or navigate confusing element family dropdowns. Instead, the WebCAE compiler features a 4-tier automated element dispatch engine that determines the optimal element formulation based on geometry dimension, meshing algorithm, interpolation order, and physics regime.

[1. Geometry & Section]  ──>  Wire: Beam (#1/#12)  |  Sheet: Shell (#2/#7/#13)  |  Solid: (#3..#6, #8..#11)
                                      │
[2. Meshing Algorithm]   ──>  Free Tet (#4/#6)     |  Hex Sweep (#3/#8/#11)     |  Inflation (#9)  |  Bridge (#10)
                                      │
[3. Interpolation Order] ──>  Order=1 (P1)        |  Order=2 (P2)               |  Sphere of Influence (Hybrid)
                                      │
[4. Physics & Solver]    ──>  Linear Statics      |  NLGEOM SE(3)/TL            |  Modal/Buckling  |  SU2 CFD FVM

The 4 Compiler Levels

1. Geometry Dimension & Structural Intent: - 1D CAD curves, centerline skeletons, or beam frames (frame, assign beam) $\to$ Element #1 (STRUCT_BEAM) or #12 (STRUCT_BEAM_EXT). - 2D CAD sheets, thin-walled plates, or mid-surfaces (shell, frame patch) $\to$ Element #2 (STRUCT_SHELL_QUAD), #7 (STRUCT_SHELL_TRI), or #13 (STRUCT_SHELL_QUAD_L). - 3D solid CAD volumes (STEP/IGES/B-Rep primitives) $\to$ Solid family (#3..#6, #8..#11). 2. Meshing Algorithm & Topological Strategy: - Free Delaunay/Frontal meshing (mesh tet) $\to$ Tet4 (#4) or Tet10 (#6). - Structured sweeping / extrusion (mesh hex) $\to$ Hex8 (#3), Hex20 (#8), or Hex27 (#11). - Prismatic wall extrusion (inflation) $\to$ Wedge6 (#9). - Hex-to-tet transition boundaries $\to$ Pyr5 (#10). - Regular Cartesian voxel grids $\to$ Voxel8 (#5). 3. Interpolation Order ($p$-Refinement): - order=1 selects linear elements (Tet4, Hex8, Wedge6, Pyr5). - order=2 selects quadratic elements with midside nodes (Tet10, Hex20, Hex27). - Local refinement zones (Sphere of Influence) elevate order locally to quadratic inside critical stress zones while keeping linear elements in the far field. 4. Physics Regime & Material Nonlinearity: - Linear statics (solve) uses standard linear small-strain operators. - Large deflection (nlgeom) activates $SE(3)$ corotational operators for beams/shells and Total Lagrangian Green–Lagrange formulations for solids. - CFD analysis (cfd) automatically converts the structural mesh into dual control volume polyhedra for SU2.

Compiler Dispatch: Base vs. Extended Formulations (1 vs 12 and 2 vs 13)

A core architectural advantage of WebCAE is automatic promotion between compact and extended formulations:

• Beams: `#1 STRUCT_BEAM` vs. `#12 STRUCT_BEAM_EXT` (governed by needsBeamExt / hasBeamExtStage3): - Base Element (`#1`): When a beam has a standard doubly-symmetric profile (round tube, rectangular box, symmetric I-beam) aligned with default axes, with zero eccentricity ($e = 0$), zero roll angle ($\beta = 0$), coincident shear center and centroid ($(y_s, z_s) = (0, 0)$), and no non-structural mass ($NSM = 0$), the compiler emits compact type 1 (STRUCT_BEAM, 6 numbers). This maximizes memory bandwidth and solver performance. - Extended Promotion (`#12`): As soon as a beam member possesses *any* extended attribute: 1. Flange eccentricity or stiffener offset: flush=outer, flush=inner, or ecc=[ey, ez] 2. Asymmetric or open-walled cross-section with shear center offset: $(y_s, z_s) \ne 0$ (e.g., channels, angles, Z-purlins) 3. Product moment of area: $I_{yz} \ne 0$ 4. Principal axis roll angle: rollDeg $\ne 0$ 5. Custom orientation vector zref / vref 6. Exact transverse shear coefficients $k_{sy}, k_{sz}$ 7. Non-structural line mass nsm $> 0$ the compiler automatically upgrades the element to `12` (`STRUCT_BEAM_EXT`, 22 numbers in `elemMat`).
• Shells: `#2 STRUCT_SHELL_QUAD` vs. `#13 STRUCT_SHELL_QUAD_L`: - Base Element (`#2`): A single-layer homogeneous isotropic sheet with no midsurface offset ($z_0 = 0$) and no non-structural mass receives compact type 2 (STRUCT_SHELL_QUAD, 3 material numbers [E, nu, t]). - Extended Promotion (`#13`): Multi-layered composite laminates (CLT / carbon ply layups), orthotropic materials, shells with an offset reference surface ($z_0 \ne 0$, which introduces extension-bending coupling matrix $\mathbf{B}$), non-structural mass ($NSM > 0$), or through-thickness thermal gradients automatically compile into `13` (`STRUCT_SHELL_QUAD_L`, 36 ABD constitutive numbers).

Compiler Mapping Reference

Local Zones: Sphere of Influence, Cylinder, Hole Collar & Hybrid Transitions

Industrial structural parts rarely require uniform mesh density. Uniform refinement across an entire assembly causes degrees of freedom to explode exponentially without improving accuracy in uncritical regions. WebCAE provides local refinement primitives that combine spatial size scaling ($h$-refinement) with local polynomial order elevation ($p$-refinement).

Sphere of Influence: Combined $h$- and $p$-Refinement

The Sphere of Influence primitive defines a spherical zone centered at coordinates $(x_c, y_c, z_c)$ with radius $R$:

mesh tet size=10
mesh refine sphere center=[50, 20, 0] radius=15 size=2 order=2

1. Spatial Size Refinement ($h$): Elements inside the sphere are refined from the baseline $10\text{ mm}$ down to $2\text{ mm}$, with smooth graded element size transitions across the sphere boundary. 2. Local Order Elevation ($p$): Inside the sphere, elements are automatically elevated to quadratic Tet10 (#6), capturing high stress gradients around fillets and notches. Outside the sphere, elements remain linear Tet4 (#4), conserving system memory. 3. Buffer Layer & LinearConstraint3 Tying: In classical FEA codes, mixing linear and quadratic elements results in "hanging nodes" at midside locations, producing displacement discontinuities and spurious stress singularities. WebCAE solves this via an automatic buffer layer and exact 3-term linear constraint equations (LinearConstraint3): $$u_m = \frac{1}{2} u_a + \frac{1}{2} u_b$$ where $u_m$ is the degree of freedom of the midside node on the interface boundary, and $u_a, u_b$ are the degrees of freedom of the corner vertex nodes of the adjacent linear element. These equations are eliminated directly during sparse matrix assembly via static condensation, guaranteeing $C^0$ displacement continuity and zero spurious stress spikes.

Cylindrical & Annular Zones: Hole Collar & O-Grid

Stress concentrations around holes in plates and shafts follow the Kirsch analytical solution ($K_t \approx 3$). Polygonal approximation of the circular boundary introduces geometric discretization errors that distort the stress peak.

1. Polar Hexahedral Hole Collar (`Hex8` / `Hex20`): Around cylindrical holes, WebCAE constructs a structured annular ring of hexahedral elements. Radial element spacing follows a geometric progression ($r_{i+1} = r_i \cdot q$), placing the finest element layer directly on the notch boundary. For quadratic meshes (Hex20), midside nodes are snapped to the analytical CAD arc, eliminating chordal facet errors. 2. 5-Block O-Grid Topology: Solid cylindrical shafts and pipes are meshed using an O-Grid block topology: a central square Cartesian core surrounded by 4 curved polar blocks. This avoids the central coordinate singularity of polar coordinates and ensures orthogonal cell angles ($90^\circ$) at the outer perimeter. 3. Conformal Hex-to-Tet Transition via Kuhn Subdivision: When a structured hex collar connects to an unstructured tetrahedral core, the hex boundary faces can be subdivided conformally into 6 tetrahedra using Kuhn's triangulation without introducing hanging nodes. Alternatively, boolean subdomains are stitched using non-conformal Mortar contact conditions.

Prisms (#9 Wedge6) and Pyramids (#10 Pyr5) in Hybrid Meshes

Hybrid meshes combine the boundary-layer resolution of hexahedra and prisms with the geometric flexibility of tetrahedra.

• Swept Prisms (`#9 Wedge6`): When sweeping a surface with mixed quad-tri topology along a guide path, quadrilateral surface facets produce Hex8 bricks, while triangular surface facets extrude into Wedge6 prisms.
• Boundary Layer Inflation: In aerodynamic and thermal skin-friction boundary layers, triangular surface meshes are extruded along surface normal vectors, forming stacks of thin Wedge6 prisms ($y^+ \approx 1$).
• Pyramidal Bridge (`#10 Pyr5`): A quad face from a Hex8 or Wedge6 cannot connect directly to a triangular face of a Tet4. Pyr5 acts as the universal conformal connector: its quadrilateral base locks onto the hex face, while its four triangular lateral faces mate with surrounding tetrahedra.

Coupled Beam-Shell & Stiffened Structures

Thin-walled structures in aerospace (fuselage skins and stringers), marine engineering (ship hulls, bulkheads, and web frames), and civil engineering (diagrid towers) rely on coupled beam-shell assemblies.

Conformal Node Coupling vs. Multi-Point Constraints

WebCAE offers two complementary workflows for coupled beam-shell modeling:

1. Parametric Frame & Shell (`frame patch`, `frame stiffen`): Generates mid-surface shell elements (MITC4) and 1D beam elements (Timoshenko) that share identical finite element node numbers along boundary and interior grid lines. No contact search or multi-point constraints are required; stiffness terms add directly into the global system matrix $\mathbf{K}$. 2. CAD B-Rep Imprinting (`imprint`, `assign beam`): When importing native STEP/IGES sheet bodies, stiffener trajectories are projected onto the B-Rep surface via imprint. The surface mesher meshes the surface conformally to the imprinted curves. Beam elements are assigned directly to the imprinted edges via assign beam edges=....

Steiner Parallel-Axis Offset Kinematics

When a beam stiffener is attached to a thin sheet, the beam centroid does not coincide with the shell midsurface:

assign beam edges=4,5,6 section=rect w=15 h=45 flush=outer

• Kinematic Offset Transformation: The displacement vector $\mathbf{u}_b$ and rotation vector $\boldsymbol{ heta}_b$ at the beam centroid are related to the shell midsurface node DOFs $\mathbf{u}_s, \boldsymbol{ heta}_s$ by: $$\mathbf{u}_b = \mathbf{u}_s + \boldsymbol{ heta}_s \times \mathbf{e}, \quad \boldsymbol{ heta}_b = \boldsymbol{ heta}_s$$ where $\mathbf{e} = [0, e_y, e_z]^T$ is the eccentricity vector.
• Transformed Stiffness Contribution: The element stiffness matrix in shell coordinates is obtained via the kinematic congruence transformation: $$\mathbf{K}_{e,\text{shell}} = \mathbf{T}_{\text{ecc}}^T \mathbf{K}_{e,\text{beam}} \mathbf{T}_{\text{ecc}}$$ This automatically couples axial extension of the beam to out-of-plane bending of the shell, producing the correct effective bending inertia: $$I_{\text{eff}} = I_s + I_b + A_{\text{eff}} e^2$$

Nonlinear Formulations, Mortar Contact & Verification Matrix

Geometrically Nonlinear Formulations (nlgeom)

WebCAE supports large deflections, large rotations, and moderate strains via two distinct kinematic frameworks:

1. $SE(3)$ Corotational Framework (Beams and Shells): For slender structural members (STRUCT_BEAM, STRUCT_SHELL_TRI, STRUCT_SHELL_QUAD), total motion is partitioned into a continuous rigid-body translation/rotation of an attached local reference frame and pure deformational strain. This completely eliminates artificial membrane strain stiffening under large rigid rotations. 2. Total Lagrangian Formulation (3D Solid Elements): For solid elements (STRUCT_SOLID_HEX, STRUCT_SOLID_TET, STRUCT_SOLID_TET10), kinematics are tracked from the initial undeformed configuration $\Omega_0$. The strain energy conjugate pair consists of the Green–Lagrange strain tensor $\mathbf{E}$ and the second Piola–Kirchhoff stress tensor $\mathbf{S}$, supporting hyperelastic material models (compressible Neo-Hookean).

Dual Mortar Contact Formulation

For assemblies with non-conformal mesh interfaces, WebCAE employs the segment-to-segment Mortar contact algorithm:

• Dual Lagrange Multipliers: Contact traction $\boldsymbol{\lambda}$ is discretized on the slave surface using dual shape functions satisfying biorthogonality: $$\int_{\Gamma_s} \Phi_i N_j d\Gamma = D_{ii} \delta_{ij}$$ This decouples the contact constraint equations, allowing the Lagrange multipliers to be condensed out of the global system without increasing bandwidth or causing zero pivots.
• Conditions: Supports frictionless unilateral contact, bonded tie constraints, and Coulomb friction with stick-slip transitions.

Element Verification Matrix

Every element in the WebCAE library is continuously tested against analytical closed-form solutions and industry-standard benchmark suites (NAFEMS, MacNeal–Harder). Full test results and deviation percentages are published openly in the Verification Suite.