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Haptics and Virtual Reality

M. Zareinejad

Lecture 10:Deformable Object

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Deformable models

Heuristic approaches

Deformable splines

Spring-mass models

Linked volume

tensor-mass model

Hybrid models

Tensor pre-computation(tensor mass)

continuum mechanical approach

Fast finite elements (FFE)

Deformable models

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Deformable splines splines

Bezier

NURBS(non-uniform rational B-splines)

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Deformable splines

Not physically-based!!

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Linked volume

Basic model

volumetric extension of the basic spring-mass model

Idea:

◦ Discrete model of deformable object = 3D chain.◦ A moving link pushes/pulls its neighboring links.◦ When the connection to a neighboring link is stretched/compressed to the

limit, motion is transferred to the respective neighbor. ◦ Small displacements of a single node cause:

Local deformations within a relaxed medium. Global deformations within a fully stressed medium.

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ChainMail algorithm

Linked volume

Deformation constraints:◦ Restrict where chain links can be relative

to their neighbors.

Stretching, compression:◦ Links within: [dxmin, dxmax]; [dymin, dymax];

and [dzmin, dzmax] of left/right, in front/behind, and top/bottom neighbors, respectively.

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ChainMail deformation constraints

zmax z

minymin

ymax x

minxmax

Requires elastic relaxation:◦ Potential energy not minimized.◦ Adjust links to satisfy energy constraints when there is time.◦ Energy defined in terms of link position w.r.t. its neighbors.◦ Object deforms approximately, and changes shape over time.

Simple to implement.

Computationally efficient/memory intensive.

Can simulate non-homogeneous & anisotropic objects.

Not physically-based:◦ Biomechanics properties -> deformation constraints?

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ChainMail – conclusions

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Finite element Model

The main idea is to use finite element method to model the mechanical properties of the relationship between stress and strain.

mechanical properties such as modulus of elasticity and Poisson ratio

Displacement any arbitrary point of the element related to the displacement of nodes, This is done with functions shape.

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Finite element Model

Static :

Dynamic :

Two types: linear and non-linear Linear : Static and Dynamic

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Finite element Model

2D:

3D:

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Finite element Model

Two-dimensional triangular element:

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Finite element Model

Pascal's triangle in two-dimensional :

Displacement any arbitrary point of the element related to the displacement of nodes, This is done with functions shape.

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Finite element Model

Static :

Stiffness matrix for each element

Strain-Displacement matrix

Material constants matrix

Since these matrices are constant for a specific element, the integral is performed on the area of each element and Stiffness matrices are computed according to the equation :

Area of each triangular element

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Finite element Model

Material constants matrix: (plane strain)

Young’s modulus

Poisson’s ratio

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Finite element Model

Strain-Displacement matrix :

Hm is the element shape function matrix

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Finite element Model

hk is the shape function of each node of the element, for the linear triangle element, this function is given by:

by substituting hk in previous equation :

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Finite element Model

: The area of each triangle is calculated as follows :

where x and y are the coordinates of nodes 1,2 and 3 of each element

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Dynamic :

Stiffness matrix for each element :

Finite element Model

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Mass matrix : Element shape function matrix

Mass density of element

Finite element Model

Damping matrix :

Damping property of element

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The position, velocity and acceleration of each node in the discretized object are updated at each simulation time step. The selection of the magnitude of this time step is important because a very

large value of time step could cause the result to diverge, while a .small value would unnecessarily increase the computations

There are two methods to solve the above equation :

Explicit Numerical IntegrationImplicit Numerical Integration

Finite element Model

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Difference formulas for approximating the velocity and accelerationin terms of displacements can be derived as follows:

Finite element Model

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By substituting previous equations into the system of dynamic equations the following relation is obtained:

Finite element Model

Easy to incorporate biomechanical properties:◦ Young’s modulus & Poisson’s ratio are included in the simulation

regardless of mesh topology.

Time step does not restrict model stiffness in dynamic simulation if semi-implicit integration is used:◦ Remeshing does not affect stability.

Suitable for describing fluids.

Performance and accuracy limits are well known.

Mesh compatible with data structures used for graphic rendering.

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FEM models - advantages

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FEM model - limitations Preprocessing:

◦ Automatic generation of good meshes is difficult.

Large deformations:◦ Remeshing.

Topology changes – cutting, tearing, fracture:◦ Remeshing.

Computational performance:◦ Interactivity remains challenging.◦ Optimizations available only for

linear problems.

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Commercial Software for FEM

ABAQUSADINAANSYSDYNA3DFEMLABGT STRUDLIDEASNASTRAN

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Real-time FEM• Parallelization• Tessellation of the problem