Mesh Format¶

HOPR HDF5 curved mesh format used by PyHOPE

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The HOPR HDF5 mesh format is designed for fast, scalable, parallel I/O of high-order unstructured 3D meshes. Mesh data is organized in non-overlapping arrays and packaged per element to minimize synchronization. A space-filling curve or structured ordering enables simple domain decomposition.

The following spectral element solvers have (optional) support for meshes generated in PyHOPE.

Framework	Language	Equation System	Reference
FLEXI	Fortran	NSE	Krais et al., 2021
ƎLEXI	Fortran	NSE/MRG	Kopper et al., 2023
GALÆXI	Fortran/C	NSE	Kurz et al., 2025
FLUXO	Fortran	NSE/MHD/Maxwell	Rueda-Ramirez et al., 2017
HORSES3D	Fortran	NSE/Cahn-Hilliard	Ferrer et al., 2023
PICLas	Fortran	Maxwell/Poisson	Fasoulas et al., 2019

Global Attributes¶

Attribute	Type	Description
Version	REAL	Mesh file format version
Ngeo	INTEGER	Polynomial degree of the curved element mapping
nElems	INTEGER	Total number of elements
nSides	INTEGER	Total number of sides (element faces)
nNodes	INTEGER	Total number of nodes
nUniqueSides	INTEGER	Total number of geometrically unique sides
nUniqueNodes	INTEGER	Total number of geometrically unique nodes
nBCs	INTEGER	Number of entries in the boundary-condition list
FEMconnect	ON/OFF	ON if FEM edge/vertex connectivity is present in the file

Data Arrays¶

Array	Type	Size	Description
ElemInfo	INTEGER	(1:6, 1:nElems)	Per-element data containing element type, zone, and offsets into side and node arrays: (ElemType, Zone, offsetIndSIDE, lastIndSIDE, offsetIndNODE, lastIndNODE).
SideInfo	INTEGER	(1:6, 1:nSides)	Per-side data stored contiguously per element range from ElemInfo. Fields: (SideType, GlobalSideID, nbElemID, 10*nbLocSide+Flip, BCID, [ElemID,locSideID]). Flip encodes side orientation; GlobalSideID<0 marks slave side.
NodeCoords	REAL	(1:3, 1:nNodes)	Node coordinates, stored per element range from ElemInfo. High-order nodes included.
GlobalNodeIDs	INTEGER	(1:nNodes)	Globally unique node IDs aligned with NodeCoords
BCNames	STRING	(1:nBCs)	List of user-defined boundary-condition names
BCType	INTEGER	(1:4, 1:nBCs)	Four-integer code per boundary condition; see boundary conditions below

Element Definitions¶

Elements are defined using non-unique node IDs stored in ElemInfo and NodeCoords. Each element has a type code and a zone ID. The zone ID can be used to group elements into physical zones or blocks.

Element Types¶

The element encoding follows CGNS-inspired conventions. The last digit of the surface type corresponds to corner count; 3D element codes distinguish linear, bilinear, and non-linear variants.

Element Type	Index	Element Type	Index	Element Type	Index
Tetrahedron, linear	104	Tetrahedron, bilinear	114	Tetrahedron, curved	204
Pyramid, linear	105	Pyramid, bilinear	115	Pyramid, curved	205
Prism/wedge, linear	106	Prism/wedge, bilinear	116	Prism/wedge, curved	206
Hexahedron, linear	108	Hexahedron, bilinear	118	Hexahedron, curved	208

High-Order Nodes¶

High-order nodes are stored in tensor-product style using (i,j,k) with uniform spacing in reference space \([-1, 1]^3\). Note that for NGeo=1, this node ordering differs from CGNS corner ordering for pyramids and hexahedra as the nodes 3/4 and 7/8 are swapped. The number of nodes per element depends on NGeo and element type.

Tetrahedron:

\((Ngeo+1)(Ngeo+2)(Ngeo+3)/6\)

Pyramid:

\((Ngeo+1)(Ngeo+2)(2Ngeo+3)/6\)

Prism/Wedge:

\((Ngeo+1)^2(Ngeo+2)/2\)

Hexahedron:

\((Ngeo+1)^3\)

Boundary Conditions¶

BCNames and BCType define the available boundary conditions. BCID in SideInfo references a 1-based index into these arrays and uses 0 for interior sides. BCType = (BoundaryType, CurveIndex, StateIndex, PeriodicIndex) contains four integers per boundary condition:

BoundaryType:

Integer code for the BC kind. Reserved values: 1 = periodic; 100 = inner/analyze sides. Periodic and inner sides also have neighbor links defined.

CurveIndex:

Geometry/CAD tag to distinguish BCs or trigger curving behavior. Currently unused in PyHOPE.

BoundaryState:

User-defined index for solver reference states; not interpreted by the format.

PeriodicIndex:

Used only for periodic sides; two matching BCs must share the same absolute value, with opposite signs.

FEM Connectivity¶

PyHOPE can optionally store additional connectivity information required by Finite Element Method (FEM)-based solvers. This includes topological connectivity of edges and vertices, as well as their connections across elements. This information is included when the global attribute FEMconnect is set to ON.

Global Attributes¶

Attribute	Type	Description
nEdges	INTEGER	Total number of entries in EdgeInfo
nVertices	INTEGER	Total number of entries in VertexInfo
nUniqueEdges	INTEGER	Total number of geometrically unique edges
nFEMSides	INTEGER	Number of topologically (incl. periodicity) unique sides
nFEMEdges	INTEGER	Number of topologically unique edges
nFEMEdgeConnections	INTEGER	Size of EdgeConnectInfo
nFEMVertices	INTEGER	Number of topologically unique vertices
nFEMVertexConnections	INTEGER	Size of VertexConnectInfo

Data Arrays¶

Array	Type	Size	Description
FEMElemInfo	INTEGER	(1:4, 1:nElems)	Per-element offsets into EdgeInfo and VertexInfo: (offsetIndEDGE, lastIndEDGE, offsetIndVERTEX, lastIndVERTEX).
EdgeInfo	INTEGER	(1:3, 1:nEdges)	For each local element edge (CGNS order): (±FEMEdgeID, offsetIndEDGEConnect, lastIndEDGEConnect). Sign encodes local-to-global orientation.
EdgeConnectInfo	INTEGER	(1:2, 1:nFEMEdgeConns)	Edge connections: (±nbElemID, ±nbLocEdgeID). Sign on nbElemID encodes primary/replica; sign on nbLocEdgeID encodes orientation.
VertexInfo	INTEGER	(1:3, 1:nVertices)	For each local vertex (CGNS corner order): (FEMVertexID, offsetIndVERTEXConnect, lastIndVERTEXConnect).
VertexConnectInfo	INTEGER	(1:2, 1:nFEMVertexConns)	Vertex connections: (±nbElemID, nbLocVertexID). Sign encodes primary/replica.