Model warping in subspace and full space.

Uchenna Ezeobi Dept of Electrical and Computer Engineering

The University of New Mexico Albuquerque, NM 87131-0001, USA

ezeob002@unm.edu

Abstract This paper compares the real time simulation of large rotation of a deformable body in subspace and full space. The use of modal analysis due to linear strain was

used to obtain the infinitesimal deformation of the body.The integration rotation of parts at each nodes was carried out to

account for large rotation of the deformable bodies. Full space and subspace method were compared for small and

large deformation. Also the stimulation time and realism of the deformed bodies for both methods were compared to

accurately examine which was better. Cholesky direct LLt method, Cholesky direct LDLt method, Conjugate method and Sparse LU method were implemented in C++ to solve the linear equation obtained from Finite Element Analysis.

We developed position and orientation constraints that would drag and twist the model to check the realism after

large deformation. From the Experiment it showed that the subspace method was more efficient, it reduced the number of computation thereby improving the stimulation speed and

also maintained the realism of the object.

Keywordsmodel reduction, FEM, deformable objects, modal warping.

I. INTRODUCTION Every object in the universe experiences deformation by

collision or any sort of impact with the surrounding. For these objects to withstand such an impact, it is highly important to be able to stimulate the effects that deformation has on every

object. In this paper we are comparing two ways of stimulating large rotational deformation of an object, which are the subspace and full space model. Large deformation of bodies is mostly solved by greens theory; which handles the linear and nonlinear part of the deformation. However, the

nonlinear calculation of the nonlinear part is very expensive to implement, which has reduced the use of these application in

computer animation. Further studies have been made to reduce the cost of implementing the large deformation of bodies.

The main goal in designing a mechanics to study the deformation of bodies is to improve the stimulation time and as well maintain the realism of the object. Both factors compensate each other. To increase the stimulation time we need to consider only the linear part and to maintain the realism the nonlinear part has to be considered. The modal analysis based on linear strain tensor reduces the computational load but the realism of the object is relatively large when twisting and bending is applied to the object. The method to be considered in this paper overcomes the

unrealistic large change in shapes of objects. It keeps track of an infinitesimal rotation tensor of each node in an object and non-linear part is omitted during the initial step. And finally at each time step modal warping is done to the object. The subspace which only incorporates the high frequencies or dominant m columns of the modal displacement 3n x m is compared with full spaces that uses all the matrix 3n x 3n. The final step taking in this paper were to solve the matrix Ax=b using different solver to compare the best method. And finally we render the solution to compare realism and stimulation time for all method.

II. RELATED WORKS For the realism of the object, grains strain tensor would be used to model the linear and non-linear part of deformation.

But high computational of the nonlinear part makes this method too expensive to implement. Linear modal analysis is another method being employed to increase the speed of the

stimulation. However, it only models the linear parts and does not account for the rotation of the objects. Due to these large

artifacts are seen when large rotation are applied to the objects.

III. BACKGROUND

These section introduces the basic principles and equations needed to stimulate the deformation of objects.

A. Kinematics of infinitesimal Deformation In other to study the deformation of an object, we have to understand the basic kinematics and formula involved in

analyzing the stimulation of the object. The final infinitesimal deformation formula is shown below:

da = dx +dx +w dx (1)

Where dx represents the strain and w dx represents the

rotation. For full derivation refer to Choi and Ko [1]. From these we could deduce that the infinite deformation depends

on the rotation and strain.

B. Finite Element Analysis The Equation for the finite Element methods is shown below:

M u..+Cu

.+Ku = F (2)

where M is the mass, C is the damping, K is the stiffness Matrix and F(t) is the external forces applied the object.

u(t) is a 3n-dimensional vector that shows the displacements of n nodes from their original position. M, C and K are constructed using the finite Element method.

The Mass is constructed using the following formula:

M = (m)H (m)T

HV (m )

m

(m)dV (m) (3)

Where (m) is the mass density, H (m) is the displacement interpolation matrix and dV(m) is the change in volume. The choice of H (m) depends on the element

geometry, number of element nodes/degree of freedom and the convergence requirements. For more details refer to [4]. And then the C is constructed using the following

equation

C = (m)H (m)T

HV (m )

m

(m)dV (m) (4)

Where k(m) is the damping property parameter of the element m. But in practice this is highly impossible to

compute, therefore C was obtained using the following: C = M +K

Where and are scalar factors. Finally the K matrices is constructed using the following

formula:

K = B(m)T

C (m)BV (m )

m

(m)dV (m) (5)

Where B(m) is the strain-displacement matrix. B(m) is

obtained by differentiating and combining rows of H(m). For more details refer to [2].

C. Modal displacements If the M, K are not diagonal Matrix, then they could be

diagonalized using the following ODE equation:

u(t) =q(t) (6) where is the modal displacement matrix and q(t) is the

vector containing all the modal amplitude. Substitute (5) into (1) yields the following equation:

Mq q..+Cq q

.+Kqq =

TF (7)

For more details refer to [1].

D. Modal Rotation The governing equation for rotation is shown below:

w(t) =Wq(t) =q(t) (8) where is the modal rotation matrix. Both the modal

displacement matrix and modal rotation matrix are only used for small deformation.

E. Large deformation To accommodate for large deformation in K(u) is replaced

with K in eqn(1). The finally equation for the method is shown below:

M uL..

+CuL.

+KuL = RTF For full derivation refer to [1]. The difference between (1) and (2) is that the external force acting on each node needs to be

rotated about its co-ordinate frame. Then position and orientation constraints were added to drag and twist the object.

Refer to [1] This equation is solved using the New mark method; so the

final equation breaks down to the following:

M + t2C + t

2

4K

"

#$

%

&'an+1 = f C(vn +

t2an )K(Un +tvn +

t2

4an )

Where the following equation has the form of Ax=b. Finally these matrix is solved using four sparse solver to compare

which solver is faster.

IV. FLOW CHART

Design

Construct Mass, Stiffness and Damping Matrix using FEM

Solve Matrix for + + = Using Sparse Solver

Integrate ! + ! + ! =

Render Mesh

V. ALGORITHM The following algorithm were used to solver the linear system equation Ax=b. These methods would be compared to check which works best for the solving a linear equation problem at

a fastest rate.

A. Sparse cholesky direct LLT factorization method This method is used to solve any linear equation in the form of Ax=b. The A matrix should be sparse positive definite matrix. For any A there is a lower triangular matrix L such that A=LLT. The pseudo-code is shown below. CHOLESKY LLT ALGORITGHM PSUEDOCODE For k=1:n

Akk= Akk for i=k+1 to n

Aik= Aik/ Akk end for j=k+1 to n

for i=j:n Aij= Aij AikAjk

end end

end

B. Sparse cholesky direct LDLT factorization method This Algorithm is similar to LLT Algorithm expect that we introduced a positive definite D. The pseudo code is shown below: CHOLESKY LDLT ALGORITGHM PSUEDOCODE Integer i,j,n,v; real array (Aij)1:n x 1:n, (lij) 1:n x 1:n, (di)1:n For j=1 to n

lij = 1

di = Ajj - dvl2jv

v=1

j1

for i= j+1 to n

lji =0

lji = Aij livdvl jvv=1

j1

#

$%

&

'( / dj

end for end for

C. Conjugate Gradient This is general method for solving Ax=b. for more details refer to [3]

D. Sparse LU Solver This method of LU decomposition is similar to Gaussian Elimination. It factors the original matrix into lower and upper diagonal matrix. This implies that Ax=Lux=b, which then turns out to be Ux=b. The pseudo code is shown below.

LU ALGORITGHM PSUEDOCODE for j = 1,, n

for i = 1,, j t=Aij if i> 1 then

for k = 1, , i-1 t= t Aik*Akj Aij = t

for i = j+1, , n t=Aij for k=1, , j-1 t=t Ajk*Akj Aij = t/Ajj

VI. EXPERIMENT

A. Data Set The data set used for this project was obtained from the subspace and full space implementation. The full space method has 3n x 3n matrix for , while the subspace

contains only the dominant columns of 3n x m.All the code for collecting the data were done with Matlab.

B. Method Comparisons The subspace and full space are two methods that could be

used to stimulate the deformation of an object. The stimulation time and realism of object of these two methods were compared. Also four different sparse solver would be

compared to see which method that would work better.

C. Preliminary result I. Subspace Method: Realism of the elephant model for the subspace is shown

FIGURE1: SUBSPACE MODEL (From Steven)

TABLE I. COMPARISON OF TIME FOR SUB SPACE

TYPE OF SOLVER TIME OF STIMULATION(SEC)

SPARSE CHOLESKY DIRECT LLT 39.91

FACTORIZATION METHOD[5]

SPARSE CHOLESKY DIRECT LDLT FACTORIZATION METHOD[5]

37.5077

CONJUGATE GRADIENT[5] 37.7418

SPARSE LU SOLVER[5] 24.32

II. Full space Method: The Realism of the object for the full space is shown below:

FIGURE 2: SUBSPACE MODEL

TABLE II. COMPARISON OF TIME FULL SPACE

TYPE OF SOLVER TIME OF STIMULATION(MIN)

SPARSE CHOLESKY DIRECT LLT FACTORIZATION METHOD[5]

1.25

SPARSE CHOLESKY DIRECT LDLT FACTORIZATION[5]

1.15

CONJUGATE GRADIENT[5] 1.18

SPARSE LU SOLVER[5] 0.75

FUTURE WORK FOR BOTH METHOD Both methods do not preserve the volume of the object when a great amount of force is applied to each node. The diagram is

shown below.

Figure 3: Volume of object not preserved (From Steven)

Figure 4: Volume of object not preserved

CONCLUSION This method is an efficient way of stimulating the real time deformation of an object but they do not preserve the volume of the object.

ACKNOWLEDGMENT The author would like to thank Prof. Mario Pattichis, Dr. Yin Yang and Steven Garcia for the motivation and

encouragement provided through the course of these project.

REFERENCES

[1] G. Min Gyu Choi and Hyeong-Seok Ko, Modal Warping: Real-Time Simulation of Large Rotational Deformation and Manipulation. IEEE Trans. ON Vis. Comp. Graph.

[2] M. Muller ,J. Dorsey,L.McMillan, R. jagnow, and B. Culter. Stable real-time deformations. In Proc. ACM SIGGRAPH Symp. Computer Animation 2002,pg. 49-54 2002.

[3] Jorge Nocedal, Stephen J. Wright, Numerical Optimization, Second ed. , Springer, 2006.

[4] Thomas J. R. Hughes, Finite Element Method, First ed. , Dover Publications,Inc. Mineola, NewYork , 2000.

[5] Eigen Library, Sparse Solver[Online document] March 26 2008, [2014 December 05], Available at FTP: http://eigen.tuxfamily.org/dox/group__TopicSparseSystems.html