Method for Graphical Display of Three-Dimensional Vector Fields

A common data form in science and engineering is the three-dimensional vector field. This data may represent information such as velocity of fluid flow or displacement of points in a continuum. A challenge facing the engineer or scientist is to efficiently develop an understanding of the data. This understanding is usually developed through the aid of visual representations such as computer graphics. Vector fields implicitly contain a large amount of data not directly nor easily observable. One of the more important factors is object deformation including local effects due to normal and shear strain and rotation. In addition it is useful to understand paths of particles in the field as well as the speed that the particles take in the vector field.

The simplest means of representing vector data is to draw an oriented and scaled line in the direction of the vector. In two dimensions this technique works reasonably well, but in three dimensions the resulting visual image is impossible to decipher correctly.

A technique referred to as streamlines uses paths that are everywhere tangent to the vector field, and are often thought of as representing the path that a massless particle would take in the fluid. Another technique called domain deformation represents the vector field by distorting the local geometry according to the vector data. These techniques fail to provide an understanding of the local deformations that exist within the vector field.

According to the present invention, a method for visualizing a three-dimensional vector field includes generating, on a graphics workstation, stream polygons and deforming each one according to components of local strain and rotation along a streamline. The radius of the stream polygons are varied, along the streamline, proportionally with a scalar function. The stream polygons can be swept along a streamline to form a stream tube. As an additional means of displaying data, the sides of the stream tube can be colored with different colors to represent some other scalar functions along the streamline. Instead of sweeping the stream polygons to create a stream tube, the stream polygons may alternatively be placed at selected intervals along a streamline.

While the novel features of the invention are set forth with particularity in the appended claims, the invention, both as to organization and content, will be better understood and appreciated, along with other objects and features thereof, from the following detailed description taken in conjunction with the drawings, in which:

FIG. 1 shows a stream tube;

FIG. 2 shows stream polygons placed along a streamline;

FIGS. 3a, 3b, 3c and 3d show stream polygons deformed according to the present invention;

FIGS. 4a and 4b show a stream polygon.

FIG. 1 shows a stream tube 10 which is a graphical device used to visualize certain aspects of a 3-dimensional vector field. The stream tube 10 is constructed mathematically by sweeping n-sided polygon 20 along a streamline 30. The streamline 30 is a path that a massless particle would take in the vector field (i.e., fluid flow) given a specified starting point for the particle. The stream polygons 20 can be rotated and deformed as they are swept to represent local strain and rotation along the streamline. The radius of the tube 10 can be varied as well, to represent changes in velocity magnitude. Each side of the tube 10 can be colored with a different (or same) color to represent additional variables simultaneously. It can be seen that by using this visualization technique, much information can be shown in a straightforward and highly compact manner.

Stream tube 10 can be constructed graphically using stream ribbons offset by a specified radius and angle from streamline 30. The dotted lines in FIG. 1 are the edges of two of the stream ribbons used in the case where the stream polygon is a quadrilateral. The ribbons when joined together, form tube 10 having the cross section of polygons 20. The offset radius of the ribbon varies with the magnitude of the vector. The rotation of stream tube 10 is achieved by rotating each of the ribbons composing tube 10. The rotation is determined by computing the local rotation of the vector field along streamline 30, or the so-called streamwise vorticity. be represented in a simple, intuitive manner.

An alternative to generating a stream tube 10 is to place multiple instances of stream polygons along streamline 30, as shown in FIG. 2. The stream polygons 20 may represent a single component of deformation or possibly combinations of deformation mode. By allowing the user to position the stream polygons interactively, it is possible to rapidly move visually along the streamline, viewing the local strain at any point.

It is necessary to characterize the vector field including local deformation, as well as techniques for computing the local vector and derivatives at any point in the vector field.

Consider a vector field .nu. consisting of the m local vectors v=(u, v, w). To examine the local deformation due to V at the point x=(x, y, z), a first order Taylor's series expansion of v about x is constructed. Then the local deformation e.sub.ij is given by

where .epsilon..sub.ij is the local strain tensor, and .omega..sub.ij is the local rotational tensor.

It can be assumed that v represents a displacement field. For general vector fields, v may represent many possible data, hence the terms deformation, strain, and rigid body motion must be interpreted accordingly. For example, if V represents fluid flow, then these terms become deformation rate, strain rate, and velocity, respectively.

The local strain is given by ##EQU1##

The terms along the diagonal of .epsilon..sub.ij are the normal components of strain. Examining a plane oriented in the x-y plane, normal strain causes uniform deformation along the x-y axes as shown in FIG. 3a. The terms off the diagonal are the shear components of strain. Shear strain causes angular deformation as illustrated by FIG. 3b.

The local rigid body rotation is given by ##EQU2##

The rotation tensor describes the local rigid body rotation as shown in FIG. 3c. By adding the contributions of the normal and shear strain and the rotation, as shown in FIGS. 3a-3c, the total deformation can be also represented as illustrated in FIG. 3d.

Note that the local rotation tensor can also be written ##EQU3## where .epsilon..sub.ijk is the alternating tensor, and .omega. is the rotation vector ##EQU4## In fluid flow where .nu. is the velocity field, .omega. is the vorticity and represents the rate of angular velocity of the flow at a point. Another important flow parameter is stream vorticity, .OMEGA., or rotation about the local vector v given by the normalized dot product. ##EQU5##

In most applications the vector field .nu. is known only at m discrete points, and an interpolation function is required to compute the derivatives at the arbitrary point x in .nu.. The geometry can be represented as a union of many non-overlapping elements, or cells, that are simple shapes such as hexahedron or tetrahedron. It is only at the vertices, or nodes, of the cells that .nu. is known. Although many choices of interpolation function are possible, the iso-parametric formulation common to finite element analysis provides many advantages including simplicity of formulation, an abundance of prior work, common use in the analysis process, and the same formulation used regardless of element topology.

The geometry of each element is described by ##EQU6## where x.sub.i are the node points of the element, and N.sub.i are the shape functions, one per element node. The shape functions vary depending upon element topology; for a hexahedron the shape functions are given in terms of the element coordinates .epsilon.=(.epsilon.,.eta.,.zeta.) ##EQU7## where .epsilon..sub.i, .eta..sub.i, .zeta..sub.i =.+-.1 at the element nodes, and .zeta.<1 inside the element. Note that at a particular element node j we choose .epsilon..sub.j, .eta..sub.j, .zeta..sub.j such that N.sub.i =1 when i=j, N.sub.i =0 otherwise.

In an isoparametric formulation, the interpolation functions N.sub.i are the same for the element geometry as well as the nodal variable, in this case the vectors v.sub.i. Hence the vector field in the element is given as ##EQU8## The local derivatives can be computed from Equations (7), (8) and (9) as

with J the Jacobian matrix ##EQU9##

Computing the streamline is straightforward. The basic approach is to integrate the equation ##EQU10## In fluid flow, v(t) is the velocity and t is time, and the path generated can be considered the motion of a massless particle in the velocity field.

Generally the integration is performed numerically using interpolation functions such as those described previously and a numerical integration scheme such as the Euler or Runge-Kutta methods. As the integration proceeds, it is necessary to track the streamline as it moves through the cells, requiring repeated transformation from global to local coordinates. This transformation is performed by solving Equation (7) explicitly for .epsilon.(x) or using a numerical technique such as Newton's method.

Consider a regular n-sided polygon whose center is located at position x and normal to the local vector v in vector field .nu.. This polygon is called a stream polygon (sp) as shown in FIG. 4a. The radius of sp is defined as the radius r of the circumscribing circle of the polygon. The parameters r and the number of sides n (FIG. 4b) are constrained by

where r.sub.max and n.sub.max are arbitrarily chosen finite values and n.sub.max is an integer.

The stream polygon provides a number of simple, yet powerful techniques for creating graphical representations of 3D vector fields. The goal is to show the usual translational effects, as well as the effects of local strain and rotation and, in addition, scalar information.

The first technique is to deform sp according to the components of local strain and rotation. Equations 2 and 3 can be combined to yield the standard transformation matrix T containing effects due to scaling (normal strain), shear, and rotation. Then sp is deformed by transformation T.

It is also possible to project local strain onto the plane defined by the point x and the local vector v. For example, if the number of sides is n=4 (i.e., the polygon is a square), the local deformation can be represented exactly as shown in FIGS. 2a-2d. Here the contributions of the strain and rotation are immediately apparent, as well as the combination of all three. Another useful technique described by Equation 6 is to project the vorticity .omega. onto the local vector v. The resulting angle .OMEGA. is the rotation of sp above v.

Another effective use of the stream polygon is to vary the radius of the stream polygon according to some prescribed relationship with vector magnitude or other scalar function. For example, if .nu. represents the velocity of an incompressible flow with no shear, then the equation ##EQU11## represents an area of constant mass flow. Here, r.sub.max is a user specified radius at the minimum flow velocity v.sub.min.

While specific embodiments of the invention have been illustrated and described herein, it is realized that modifications and changes will occur to those skilled in the art. It is therefore to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit and scope of the invention.