Deformable Bodies
Deformation rest space deformed space x p ( x ) • 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.
• 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) r p T r p � I = ( RS ) T RS � I = S T R T RS � I = S T 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 x p ( x ) • Green’s strain • Cauchy’s strain
• Measurement of deformation • Measurement of elastic force • Constitutive law • Finite element method
Elastic force • 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
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 • σ 11 measures force normal to the plane (normal stress) • σ 21 and σ 31 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? n = { 1 2 , 1 2 , 0 } √ √ p
• Measurement of deformation • Measurement of elastic force • Constitutive law • Finite element method
Constitutive law • 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 ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 • What is the dimension of C ?
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 x 0 , x 1 , x 2 , x 3 • Deformed shape is represented by p 0 , p 1 , p 2 , p 3 • 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 / 2 Distribute f 0,1,2 evenly to p 0 , p 1 , and p 2 Repeat for other three faces
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 R T p • calculate the internal force K ( R T p − x ) • rotate it back to the deformed frame: f = RK ( R T p − 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 = [ a 0 , a 1 , a 2 ]
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