Glossary

This page contains definitions for various terminology and abbreviations used throughout opensg documentation and code.

Terminology

ABD Matrix: A 6x6 matrix used in classical lamination theory to relate generalized forces and moments to generalized strains and curvatures. The matrix has the structure [[A, B], [B, D]] where A is the 3x3 membrane stiffness, B is the 3x3 coupling stiffness, and D is the 3x3 bending stiffness.

Boundary Condition: A condition imposed on facets or boundaries of a mesh, such as Dirichlet boundary conditions (specifying function values on the boundary) or Neumann boundary conditions (specifying derivative values).

Cell: A fundamental element of a mesh (e.g., a triangle in 2D, tetrahedron in 3D) that divides the domain into discrete parts for FEA computations.

Composite Laminate: A structural material made up of multiple layers (plies) of fiber-reinforced material, each potentially having different fiber orientations and material properties.

Composite Layup: See Composite Laminate.

Connectivity: The relationship between different entities in a mesh (e.g., between vertices and cells, cells and facets). Connectivity maps define how entities are connected within a mesh.

Dehomogenization: The process of recovering local stress and strain fields from homogenized beam properties using fluctuating functions. This allows analysis of detailed stress distributions within the original 3D structure.

Eigenvalue Analysis: Mathematical technique used to determine natural frequencies, buckling loads, or other critical values by solving generalized eigenvalue problems of the form Ax = λBx.

Euler-Bernoulli Beam Theory: A classical beam theory that assumes plane sections remain plane and perpendicular to the neutral axis after deformation. Neglects shear deformation effects.

Facet: A boundary entity of a cell that is one dimension lower than the cell itself.

Fluctuating Functions: Mathematical functions that represent the deviation from average behavior in homogenization theory. Used to capture local effects within the Structure Genome.

Frame: Local coordinate system defined at each point in the mesh, typically consisting of orthogonal unit vectors (e1, e2, e3) that define material orientations.

Gamma Functions: Mathematical operators (gamma_h, gamma_e, gamma_l, gamma_d) used in MSG formulations to relate displacement fields to strain measures for different beam theories.

Homogenization: The process of computing effective structural properties (stiffness matrices, mass properties) from either 1D, 2D, or 3D SGs.

Layup: The sequence and orientation of composite layers (plies) that make up a laminated structure, including thickness, material, and fiber angle for each layer.

Local Frame: A coordinate system attached to specific geometric entities (edges, faces) that defines the orientation of materials and the direction of structural axes.

Mesh: A collection of cells (e.g., triangles, quadrilaterals in 2D, or tetrahedrons, hexahedrons in 3D) representing a discretized computational domain. A dolfinx.mesh.Mesh object encapsulates both the geometry and topology of this mesh.

Nullspace: In finite element analysis, the mathematical space containing rigid body motions that must be constrained to ensure a unique solution to the structural problem.

Segment: A portion of a blade mesh that is analyzed as a separate entity. Wind turbine blades are typically divided into multiple segments along their span for analysis.

Shell Element: A finite element type used for modeling thin-walled structures like wind turbine blade skins.

Solid Element: A finite element type used for modeling three-dimensional volumetric structures.

Stiffness Matrix: A matrix that relates applied forces to resulting displacements in a structural system. Can be 4x4 for Euler-Bernoulli beams or 6x6 for Timoshenko beams.

Structure Gene (SG): The fundamental building block in MSG theory - a representative cross-section of the structure that captures all geometric and material complexity in 1D, 2D, or 3D form.

Timoshenko Beam Theory: An enhanced beam theory that includes the effects of shear deformation and rotary inertia, providing more accurate results for thick beams or shells.

Topology: In dolfinx, refers to the structural organization of the mesh, including the number and types of entities (vertices, edges, faces, cells) and their connectivity.

Abbreviations

ABD: A-B-D matrix (membrane-bending-coupling stiffness matrix)

BECAS: BEam Cross section Analysis Software (alternative cross-sectional analysis tool)

DOF: Degrees of Freedom

DOLFINx: The next-generation finite element library (part of FEniCS)

EB: Euler-Bernoulli (beam theory)

EPS: Eigenvalue Problem Solver (from SLEPc library)

FEA: Finite Element Analysis

FEniCS: Open-source finite element computing platform

GEP: Generalized Eigenvalue Problem

GMSH: 3D finite element mesh generator

MPI: Message Passing Interface (for parallel computing)

MSG: Mechanics of Structure Genome

MSH: GMSH mesh file format

PETSc: Portable Extensible Toolkit for Scientific Computation

SG: Structure Gene

SLEPc: Scalable Library for Eigenvalue Problem Computations

UFL: Unified Form Language (part of FEniCS for expressing finite element forms)

VABS: Variational Asymptotic Beam Sectional Analysis (beam cross-section analysis tool)

YAML: Yet Another Markup Language (data serialization format used for input files)

Variable Name Conventions

Timoshenko Beam Model (Tapered Shell Wind Blade Segment)

Input: .yaml File

Output: Timoshenko Stiffness Matrix:

• Boundary Timoshenko Stiffness (Deff_l/Deff_r)

• Entire Blade Segment Timoshenko Stiffness (Deff_srt)

Byproduct and intermediate step of computation:

• Boundary Euler-Bernoulli Stiffness (D_effEB_l/D_effEB_r)

• Entire Blade Segment Euler-Bernoulli Stiffness (D_eff)

Input Variables/Data:

section_id

Constant - Segment_id of wind blade to be computed

mesh

mesh data of entire wind blade (WB) segment

subdomains

layup id of mesh elements (a.k.a. physical domain of mesh)

boundaries

facet id of boundary elements of WB mesh (Not useful in computation)

x_min, x_max

minimum/maximum x (beam axis) coordinate to define boundary mesh

tdim, fdim

mesh topology/facet dimension (For quad mesh: tdim=2, fdim=1)

facets_left/right

Left/Right boundary facet id of WB mesh (In WB numbering) - useful in generating boundary mesh

mesh_r/mesh_l

left and right boundary mesh (Generated in dolfinx as submesh)

entity_mapl/mapr

facet id of left/right boundary (In boundary mesh numbering)

num_cells

number of quadrilateral mesh elements

o_cell_idx

Original mesh element connectivity (SG_Shell.msh). Note: dolfinx does the renumbering of vertex and mesh connectivity

VV

functionspace (Discontinuous galerkin) to store orientation data

frame ([EE1,EE2,N])

Local Orientation frame (from .yaml) stored as ufl function

• [EE1,EE2,N]: EE1-> curvilinear tangent direction (along beam axis), EE2-> Circumferential tangent Direction (CCW about +x (beam) axis), N -> Inward normal direction

material_parameters

Stores anisotropic material properties of layups (Same for all WB Segments)

Layup and Material Data:

nphases

number of layups in the WB segment

nlay

[nphases,1] - number of layers in each layup of WB segment

matid

[nphases,nlay,1] - Contains thickness data of layup id- nlay

thick

[nphases,nlay,1]

conn3

[num_cells,4] - facet connectivity matrix for each element

subdomains_l

layup id of left boundary mesh (Output for subdomains.values[:] arranged in boundary dofs)

frame_l

Local orientation frame (orthogonal) for each left boundary element

• [E1l,E2l,Nl]: E1l-> beam axis (+x dir), E2l-> Circumferential tangent Direction (CCW about +x (beam) axis), E3l -> Inward normal direction

boundary_facets_left

[num_facets_left_boundary,1] - facet id connectivity of boundary mesh (In mesh_l based numbering)

D_effEB_l/D_effEB_r

Euler-Bernoulli matrix of left/right boundary of WB Segment

Deff_l/Deff_r

Timoshenko Stiffness matrix of left/right boundary of WB Segment

Function Spaces and Solution Variables:

ndofs_WBmesh

[3*len(np.arange(*V.dofmap.index_map.local_range)),1] - total dofs of WB segment mesh (Note: V=> functionspace of WB mesh)

V0_l/V0_r

[ndofs_leftmesh,4] - Fluctuating function solution after solving 4 load cases (for EB Stiffness/a.k.a zeroth order)

V1_l/V1_r

[ndofs_leftmesh,4] - Fluctuating function solution after solving 4 load cases (for Timo Stiffness/ a.k.a first order)

e_l

Local orientation Frame for left boundary mesh interpolated in V_l (Serendipidity) functionspace

e_r

Local orientation Frame for right boundary mesh interpolated in V_r (Serendipidity) functionspace

e

Local orientation Frame for WB Segment mesh interpolated in V (Serendipidity) functionspace

V_l/V_r

functionspace of left/right boundary mesh (UFL)

V

functionspace for defining WB segment mesh (UFL)

dvl, v_l

Trial and Test Functions for left boundary (V_l) (UFL)

dvr, v_r

Trial and Test Functions for right boundary (V_r) (UFL)

dv, v_

Trial and Test Functions for WB mesh (V) (UFL)

x_l, x_r

Spatial Coordinate Data for left/right boundary (UFL)

dx_l

Measure for left boundary mesh with subdomains_l assigned

dx

Measure for WB mesh with subdomains assigned (Used in defining weak form *dx(i) mean integration over mesh elements with layup id i)

nullspace_l

contain nullspace vector for used as constraint to block Rigid body motion when solving variation form (ksp solver) over left boundary mesh

ndofs_leftmesh

[3*len(np.arange(*V.dofmap.index_map.local_range)),1] - total dofs of left boundary mesh

Assembly Matrices and Weak Forms:

A_l

[ndofs_leftmesh,ndofs_leftmesh] - Global Assembly Coefficient matrix (in ufl form) for left boundary

• Can print matrix as array by A_l[:,:]

• Used in solving 4 load cases where A_l w_l = F_l (similar to Ax=b)

• Unknown w_l is stored in V0_l[:,p] for load case p

A

[ndofs_WBmesh,ndofs_WBmesh] - Global Assembly matrix for WB mesh

a

Bilinear weak form for WB mesh - a(dv,v_)

F2

Linear weak form for WB mesh - F2(v_)

• Can compare with weak form to solve as a(dv,v_) w = F2(v_), where w.vector[ndofs_WBmesh,1] is unknown dofs value stored in V0[ndofs_WBmesh,4]

F

[ndofs_WBmesh,1] - Global Assembled Right hand vector for EB case

b

[ndofs_WBmesh, 4] - Equivalent to Right hand vector F for Timo case

v0

Stores the solved V0[ndofs_WBmesh,p] in ufl function form for each load case

v2a

UFL function defined over V (functionspace_WB mesh) - Output by v2a.vector[:]

bc

Dirichlet bc (known ufl function storing boundary solutions, dofs of boundary need to be constrained)

Stiffness Matrix Components:

Dhe

[ndofs_WBmesh,4] - < gamma_h.T ABD gamma_e>

Dle

[ndofs_WBmesh,4] - < gamma_l.T ABD gamma_e>

Dll

[ndofs_WBmesh,ndofs_WBmesh] - < gamma_l.T ABD gamma_l>

Dhl

[ndofs_WBmesh,ndofs_WBmesh] - < gamma_h.T ABD gamma_l>

D_ee

[4,4] - < gamma_e.T ABD gamma_e>

D1

[4,4] - < V0.T -Dhe> Information matrix from fluctuating function data

V0

[ndofs_WBmesh, 4] - Solve [A w =F] Fluctuating function after solving 4 load cases on WB mesh (EB)

V1s

[ndofs_WBmesh, 4] - Solve [A w =b] Fluctuating function after solving 4 load cases on WB mesh (Timo)

D_eff

[4,4] - Euler-Bernoulli Stiffness matrix of entire WB segment

Deff_srt

[6,6] - Timoshenko Stiffness matrix of entire WB segment

Note

For Boundary Timo Solving, Dhe, Dle, Dll, Dhl, Dee use (dofs_leftmesh/dofs_rightmesh) in place of dofs_WBmesh