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

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• 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

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• Need to compute deformation gradient

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• Both displacement gradient and deformation gradient are translation invariant but rotation variant

Green’s strain• Green’s strain can be defined as

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• Green’s strain is rigid-motion invariant (both translation and rotation invariant)

rpTrp� 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?

• 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?

Definition of strain

• Choose a material point 0 in the rest space and express a neighborhood point in the deformed space as

Zero strain constraint

• To make sure neighborhood does not stretch, compress, or sheer

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• Therefore, strain is defined to measure

• Substituting deformation gradient with displacement gradient, we get Green’s strain

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

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• 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

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• 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

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• 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?

pn = { 1p

2,1p2, 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

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• 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?

2

4"11 "12 "13

"21 "22 "23

"31 "32 "33

3

5

Materials• For a homogeneous isotropic elastic material, two independent

parameters are enough to characterize the relationship between stress and strain

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• 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

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• 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:

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• 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

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

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• 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:

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• Use Gram-Schmidt method to approximate the closest rotation matrix to A.

where A = [a0, a1, a2]