Step 2: Create a convex polyhedron /hull of the found extreme points;

We use the incremental convex hull algorithm [20] to calculate the convex polyhedron /hull of the previously found extreme points.

After calculating the convex hull of the extreme points, it is needed to check whether each points locate inside the convex polyhedron /hull.

After forming the convex polyhedron /hull of the extreme points, those interior points that locate inside the convex polyhedron can be directly discarded.

We adopt the incremental algorithm [20] to calculate the convex polyhedron /hull of the previously found extreme points.

We also design a simple CUDA kernel to discard the interior points that locating inside the convex polyhedron /hull formed in the previous step; see the lines 42 ~ 50 in Fig.3.

4 also provides the convex polyhedron formed by the found extreme points in the use of CudaPre3D, and the desired convex hull for each set of input points derived from those 3D mesh models.

Thanks to the new techniques to calculate integrals in [??]([x.sub.t]) and the convex polyhedron method to check thenegative definite for the upper bound of V([x.sub.t]), the resulting Theorem 2 is expected to be less conservative with fewer matrix variables, as shown in the following example.

On the one hand, using the convex polyhedron method we can prove that [bar]([alpha], [beta]) < 0 by (34).

Therefore, if [[OMEGA].sub.A] is a convex polyhedron, then there exists q > 0 such that [chi] [member of] [H.sup.2+q] ([[OMEGA].sub.A]); see [23].

Therefore, when [[OMEGA].sub.A] is a convex polyhedron, it makes sense to approximate A [member of] X by finite elements from [X.sub.h].