15.3 File Formats

15.3.1 The Bézier extraction file format

15.3.1.1 Introduction

The support of novel computer-aided geometric descriptions forming a potential future basis of isogeometric analysis in ANSYS LS-DYNA is discussed in the present document. In particular, the design of a new keyword as well as the structure of the geometry input file meant to replace the current method using the *INCLUDE_TRANSFORM keyword is outlined. It is also aimed to generalize the geometric description as well as to focus on compressed storage in order to enable the run of larger and more complex examples.

The remaining part of the document is structured as follows. The new LS-DYNA keyword is introduced in ANSYS LS-DYNA keyword. Supported geometry file formats are discussed in Geometry file formats in greater depth.

15.3.1.2 ANSYS LS-DYNA keyword

The *IGA_INCLUDE_{OPTION1}_{OPTION2} keyword is introduced to import geometry files to LS-DYNA with BEZIER being one of the supported first optional arguments and blank or TRANSFORM as second optional argument, i.e.

15.3.1.3 Geometry file formats

The key differences with respect to the previous format are as follows:

1. Reduced storage: Bézier extraction operators are not stored in their full form henceforth. In what follows, we distinguish between tensor product, non-tensor-product, and mixed elements. A d-dimensional tensor product element is defined by d local knot vectors. A non-tensor-product element is defined by a set of coefficient vectors essentially representing a row in the Bézier extraction operator. Elements with mixed tensor and non-tensor-product structure, e.g. prism cf. ASCII format may be defined using a combination of local knot and coefficient vectors.

2. Sorted input: Local knot and coefficient vectors are collected into sorted blocks comprised in a library. Element definitions use local knot and/or coefficient vector identifiers, i.e. pointers to the entries of the library. Furthermore, a coefficient vector may be stored using either the dense or sparse storage formats. Noting that the latter may be beneficial as the element dimension increases but complicate the export of the data, the choice to invoke different storage formats is left to the preprocessor. Assuming, for instance, that tensor product elements are used to define most part of the discretization and non-tensor-product elements occur in the vicinity of a few extraordinary points only, it might be easier to export the data in dense format only.

3. Format and precision: In order to ensure consistency with the binary input, cf. Binary format, a fixed input format is proposed using (due to the relative indexing) short integers, i.e. $i8$, and double precision reals of the form 1PE24.16. Consequently, each line may contain up to ten integers or five reals yielding lines of up to 80 or 120 character long, respectively.

For brevity, local knot vectors are also referred to as coefficient vectors henceforth.

15.3.1.3.1 ASCII format

The following structured input has to be written for each patch separately, i.e.

15.3.1.3.2 Binary format

In addition to the ASCII format outlined in the previous section, we intend and in most industrial cases prefer to support binary storage of the geometry using the open LSDA format and API developed and maintained by LSTC. Invoking the binary format will further reduce storage requirements, speeding up I/O. The data should be written using the following path: "keyword/[option]/patch[i8.8]/" where "[option]" is either "isoshell" or "isosolid" for two and three-dimensional patches, respectively.

15.3.2 The Exodus Bézier extraction file format

This document describes a standard format for exporting Bézier extraction information in the Exodus-II file format, v8.03 and newer (simply referred to as the Exodus format for the remainder of this document). We leverage the existing attribute capability in Exodus to output the data needed for Bézier extraction, including control point weights, additional element information, and coefficient vector information. In the following sections, we establish a set of common attribute names used to tag this information.

15.3.2.1 Control point information

Control point information in Bézier extraction is always encoded in homogeneous form. Nodal values, such as spatial coordinates, are always transfered to projective space through a multiplication by their respective weights. The spline control points are written to Exodus in homogeneous form as nodal control points using ex_put_coord. The nodal weights are stored as nodal attribute, bex_weight, as described below.

In the table above, the (†) symbol indicates that the attribute is optional. If all control weights are one, as is the case for non-rational geometry, this attribute may be omitted. If no bex_weight attribute is specified, the weights are implied to be equal to 1.0.

15.3.2.2 Element information

To explicitly denote Bézier elements, they are named according to established Exodus element names with a BEX_ prefix.

These new elements and their corresponding parametric dimension d are given in the table below:

Note that the Bézier extraction format does not currently support pyramid elements.

To explicitly denote Bézier elements, they are named according to established Exodus element names with a BEX_ prefix. For each Bézier element, we store the polynomial degree of the element as an entity attribute. In table below num_deg is the length of the bex_elem_degrees vector p that uniquely defines the Bernstein polynomials over the element.

For each element block, we store three additional attributes to encode the information necessary to define Bézier elements. In the EX_ELEM_BLOCK table below, d denotes the element’s parametric dimension, and num_elem_blk denotes the number of elements in the block.

If the extraction operator is simply the identity matrix for every element in the mesh, the spline is composed entirely of $C^{0}$ Bézier elements. In this case, the extraction operators are redundant and the optional attributes indicated by the † symbol do not need to be written to the file. If these optional attributes are not included for a block of Bézier elements, it is implied that the extraction operator is simply the identity matrix for every element in the block.

15.3.2.3 Node numbering

The node numbering for the Exodus Bézier elements follows an $i$, then $j$, then $k$ fastest ordering. While this is a departure from the node ordering traditionally used in Exodus, it is necessary for easily supporting elements with arbitrary polynomial degree. Figure 604 through Figure 608 give examples of this node ordering on quadratic elements.

Figure 609 gives an example of this indexing scheme on a Bézier hexahedron with polynomial degree $p = {2,1,3}$.

Figure 604: Node ordering for a BEX_CURVE element with degree $p = 2$.

Figure 605: At left: Node ordering for a BEX_QUAD element with degree $p = {2,2}$. At right: Node ordering for a BEX_TRIANGLE element with degree $p = 2$

Figure 606: Node ordering for a BEX_HEX element with degree $p = {2,2,2}$. From left to right, the figures show the nodes corresponding to slices at $k=1$, $k=2$, and $k=3$, respectively.

Figure 607: Node ordering for a BEX_WEDGE element with degree $p = {2,2}$. From left to right, the figures show the nodes corresponding to slices at at $k=1$, $k=2$, and $k=3$, respectively.

Figure 608: Node ordering for a BEX_TETRA element with degree $p = 2$. From left to right, the figures show the nodes corresponding to slices at at at $k=1$, $k=2$, and $k=3$, respectively.

Figure 609: Node ordering for a BEX_HEX element with degree $p = {2,1,3}$. From left to right, the figures show the nodes corresponding to slices at at at $k=1$, $k=2$, and $k=3$, respectively.

15.3.2.4 Coefficient vectors

The final set of attributes that must be stored are the extraction operator coefficient vectors. Many Bézier extraction file formats allow for both dense and sparse storage of coefficient vectors. Dense storage entails storing every coefficient vector entry for the extraction operators on multivariate ($d \gt 1$) elements. Sparse storage is an optimization scheme for tensor-product elements, wherein only the coefficient vectors for univariate extraction operators are stored. The coefficient vectors for tensor product elements can then be computed from the tensor product of these univariate coefficient vectors. As an example, the coefficient vector for a degree p = {2,2} BEX_QUAD element

$$c = { 0, 0, 0, 0, 0, 0, 0, 1/2, 1/4 }$$

can be computed as the Kronecker product of two $p = 2$ univariate coefficient vectors

$$c = { 0, 0, 1/2 } \otimes { 0, 1, 1/2 }.$$

Currently, only dense coefficient vector storage has been specified for Exodus, but sparse storage will be added in the future. The coefficient vector is stored as an EX_BLOB with name bex_cv_blob Let NDCVB denote the number of dense coefficient vector blocks in the file. The attributes and variables stored on the blob are given in the table below.

Note that every coefficient vector attribute is marked as optional (†). If no coefficient vector attributes are found in a file containing Bézier elements, the extraction operator is assumed to be the identity matrix for every element. In the case of an Exodus file with transient data, the bex_dense_cv_blocks variable will always be stored at the first time step time_step=1.

15.3.2.4.1 Examples

In this final section, we supply three simple examples to illustrate how Bézier extraction information is encoded in the Exodus format.

15.3.2.4.1.1 Bézier mesh with two elements

In this first example, we save a simple Bézier mesh as an Exodus file. The mesh is a simple rectangle composed of two biquadratic elements, and the interelement continuity is one. Using Coreform Cubit, the Exodus file can be generated with the following commands:

Example journal file that creates a simple U-spline and exports the Bézier mesh Exodus format

reset reset iga create surf rect width 2 height 1 move surf 1 x 1 y 0.5 surf 1 size 1 mesh surf 1 uspline surf 1 export uspline 1 exodus ’two_elem_example’

The resulting Bézier mesh is shown in figure 610.

The tables below show the expected output data contained in the Exodus file.

Figure 610: Schematic of the simple U-spline surface created by the Coreform Cubit commands shown above. The surface is composed of two biquadratic Bézier elements, $c_{1}$ and $c_{2}$, with $C^{0}$ continuity between elements. The global control points are shown in red.

15.3.2.4.1.2 Spline mesh with two elements

In the previous example, the extraction operator was applied to the spline mesh before writing down the data to the Exodus file, so the Bézier mesh was saved into the Exodus file. In the present example, we save the spline mesh and its extraction operators directly to the Exodus file.

To generate the file, we change the last line of the Coreform Cubit command shown above in Bézier mesh with two elements by

export uspline 1 exodus ‘two_elem_example’

The resulting U-spline surface is shown in figure 611.

Figure 611: Schematic of the simple U-spline surface created by the Coreform Cubit commands shown above. The surface is composed of two biquadratic Bézier elements, $c_{1}$ and $c_{2}$, with $C^{1}$ continuity between elements. The global control points are shown in red.

15.3.2.4.1.3 Rational spline mesh with four elements

This example shows just the data entries for a four element mesh of a quarter section of a plate with a hole.

Figure 612: Quarter segment of a plate with a hole.