Visualization Algorithm

The new visualization algorithm is based on the reverse process of auto meshing. There are three steps in this algorithm: first, solve the above equations with the known and unknown reversed, then search among the big elements, and finally, compute the value with proper interpolation functions.

In a single big element, the current problem is that is known and it is necessary to find . In other words, the known and unknown are reversed in the above equations compared with meshing. It is very difficult to reverse these equations analytically, partially because the shape function N is different from one element type to another. The easiest way is to use the Newton-Raphson method to solve these equations numerically. It can produce sufficiently precise values of , and it does not care about what is the shape function.

The initial guess for is , because and . The iteration is terminated when and . This convergence condition ensures that the error is less than one unit in real regular space. In solving the equations, the Newton-Raphson method definitely converges, because the inverse of the Jacobian matrix exists, and its coefficients are the first derivatives of the shape function. In FEM analysis, this inverse matrix is also used in projection back and forth between those two coordinate systems. The rules in FEM guarantee existence of the inverse matrix [1] [2] [8]. Also, the Jacobian matrix can be analytically derived and evaluated, because N is a set of simple polynomials. The Newton-Raphson method converges quadratically, and the initial solution is very close to the solution of the equations. In numerical experiments, it only takes three or four iterations to find the solutions.

By now, a single big-element has successfully been projected to regular space. For the whole structure, the only way is to search among big-elements, because there is no requirement for how a structure is built up of big-elements. It will need a few more iterations, but usually after one iteration, it can be known if a point is inside a big-element. A structure mostly consists of dozens of big-elements. That seems to be a drawback of this visualization algorithm, but on the other hand the scale of the problem is reduced greatly. The search is now performed among dozens of items rather than among thousands. The advantage is obvious.

The next step is to transform the solution to real regular space. It is a matter of simply scaling and transforming. By the way, this method can also determine whether a given point is inside a big-element or not, because the ranges of and are already known. So it can also judge if a point is inside the volume.

Using the above method, the lower left corner point coordinates can be found in regular space for a given point. Then, the rest of the surrounding points can be very easily found by increasing one unit in every direction.

The pseudo code of this process is as follows:

 Loop for all big elements

 			 Using the Newton-Raphson method to solve the equations;

 			 		 Exit, if the point is inside the current big element;

 			 		 Continue, if outside

			 End of loop

			 If the point is inside the volume then

        		 Transform the solution  into regular space;

			 		   Return 

			  Else

        		 Return not found;

			 End of if

The value at point can be computed by interpolation with the consideration of what kind of element is used in the analysis phase, after its surrounding points are found. This step seems very trivial, but it is very important to reduce errors. It is incorrect to use linear interpolation to compute this value in all cases, because the shape function is not always a linear function. The shape function used in FEM analysis must be used as the interpolation function here, because the distribution pattern was determined when the element type was chosen in the analysis phase. Any other interpolation function can not be used. Otherwise, the distribution graphs will be very odd. At least there will be abnormal jumps (not continuous in their first derivatives) at boundaries of cells. That will give a false feedback to designers, because in a good design stress changes reasonably smoothly and stress jumps signify a serious design defect.

In this visualization process, there is no need to preprocess the whole data set, because projecting a point into regular space is very fast. Whenever a point needs to be projected into regular space , just call this function. This way, not much memory space is needed. Then a visualization method, such as ray tracing, can be used to render the whole volume.