Deformable Bodies Deformation rest space deformed space x p ( x ) - - PowerPoint PPT Presentation

Deformable Bodies

Deformation

• Given a rest shape x and its deformed configuration p(x), how

large is the internal restoring force f(p)?

• To answer this question, we need a way to measure

deformation.

x p(x) rest space deformed space

• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method

Displacement field

• Displacement field directly measures the difference between

the rest shape and the deformed shape

• It’s not rigid-motion invariant. For example, a pure translation

p = x + 1 results in nonzero displacement field u = 1

Displacement gradient

• Displacement gradient is a matrix field
• Need to compute deformation gradient
• Both displacement gradient and deformation gradient are

translation invariant but rotation variant

Green’s strain

• Green’s strain can be defined as
• Green’s strain is rigid-motion invariant (both translation and

rotation invariant)

rpT rp I = (RS)T RS I = ST RT RS I = ST S I

Cauchy’s strain

• When the deformation is small, Cauchy’s strain is a good

approximation of Green’s strain

• Is Cauchy’s strain rigid motion invariant?

Quiz

• Consider a point at rest shape x = (x, y, z)T and its deformed

shape p = (-y, x, z)T, what is the Cauchy’s strain for this deformation?

Quiz

• Deformation of an object can be measured in different ways.

Suppose a shape x undergoes a deformation to shape p(x). Please discuss whether each of the following deformation measurement is 1) translational invariant and 2) rotational invariant.

• u = p(x) - x
• del u
• Green’s strain
• Cauchy’s strain

x p(x)

• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method

• Strain measures deformation, but how do we measure elastic

force due to a deformation?

• Stress measures force per area acting on an arbitrary imaginary

plane passing through an internal point of a deformable body

• Like strain, there are many formula to measure stress, such as

Cauchy’s stress, first Piola-Kirchhoff stress, second Piola- Kirchhoff stress, etc

Elastic force

Stress

• Stress is represented as a 3 by 3 matrix, which relation to force

can be expressed as

• da is the infinitesimal area of the imaginary plane upon

which the stress acts on

• n is the outward normal of the imaginary plane.

Cauchy’s stress

• All quantities (i.e. f , da and n) are defined in deformed

configuration

• Consider this example, what is the force per area at the

rightmost plane?

Cauchy’s stress

• The internal force per area at the right most plane is
• σ11 measures force normal to the plane (normal stress)
• σ21 and σ31 measure force parallel to the plane (shear stress)

Quiz

• Given the stress matrix below around a point p, what is the

normal stress on the following surface?

p

n = { 1 √ 2, 1 √ 2, 0}

• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method

• Constitutive law is the formula that gives the mathematical

relationship between stress and strain

• In 1D, we have Hooke’s law
• Constitutive law is analogous to Hooke’s law in 3D, but it is

not as simple as it looks

Constitutive law

Constitutive law

• What is the dimension of C?

  ε11 ε12 ε13 ε21 ε22 ε23 ε31 ε32 ε33  

Materials

• For a homogeneous isotropic elastic material, two independent

parameters are enough to characterize the relationship between stress and strain

• E is the Young’s modulus, which characterize how stiff the

material is

• ν is the Poisson ratio, ranging from 0 to 0.5, which describe

whether material preserves its volume under deformation

• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method

Finite element method

• So far we view deformable body as a continuum, but in

practice we discretize it into a finite number of elements

• The elements have finite size and cover the entire domain

without overlaps

• Within each element, the vector field is described by an

analytical formula that depends on positions of vertices belonging to the element

Tetrahedron

• Rest shape of a tetrahedron is represented by x0, x1, x2, x3
• Deformed shape is represented by p0, p1, p2, p3
• Any point x inside the tetrahedron in the rest shape can be

expressed using the barycentric coordinate

Barycentric coordinates

• FEM assumes that deformed shape is linearly related to rest

shape within each tetrahedron

• Therefore, p(x) can be interpolated using the same barycentric

coordinates of x

• p(x) can also be computed as

Quiz

• What is the Green’s strain of the deformed tetrahedron?

Elastic force

• To simulate each vertex on a tetrahedra mesh, we need to

compute elastic force applied to vertex

• Based on p(x), compute current strain of each tetrahedron
• Use constitutive law to compute stress
• For each face of tetrahedron, calculate internal force:
• A is the area of the face and n is the outward face normal
• Distribute the force on each face to its vertices

Compute internal force

Repeat for other three faces Distribute f0,1,2 evenly to p0, p1, and p2 / 2

Linear assumptions

• Material linearity: The relation between strain and stress obeys

Hooke’s law.

• Geometry linearity: A linear measure of strain such as

Cauchy’s strain.

• Using these two assumption together, we can assume linear

PDE.

• In addition, we assume deformation is small around rest shape

and calculate face normal and area using rest shape.

Linear FEM

• Simplified relationship between internal force and deformation
• For one tetrahedron, K is a 12 by 12 matrix and can be pre-

computed and maintain constant over time.

• Use the assumptions in previous slide to compute internal force

for one tetrahedron, equate it with K(p - x), and solve for K.

Recipe to compute stiff matrix

Compute each 3x3 submatrix of K where,

Stiffness warping

• Because the stiffness matrix only depends on the rest shape, it

is only correct when the deformation is small.

• Catchy strain cannot capture rotational deformations correctly.

Corotational FEM

• When object undergoes rotation, the assumption of small

deformation is invalid because Cauchy’s strain is not rotation invariant

• Corotational FEM is an effective method to eliminate the

artifact due to rotation

• first extract rotation R from the deformation
• rotate the deformed tetrahedron to the unrotated frame RTp
• calculate the internal force K(RTp − x)
• rotate it back to the deformed frame: f = RK(RTp − x)

Corotational FEM

Extract rotational matrix

• Non-translational part of deformation:
• Use Gram-Schmidt method to approximate the closest rotation

matrix to A.

where A = [a0, a1, a2]