Physics Based Modeling II Deformable Bodies

Lecture 12Kwang Hee KoGwangju Institute of Science and Technology

Introduction Solving the Lagrange equation of motion

In general, it is not easy to analytically solve the equation. The situation becomes worse when a deformable body is used.

We use the finite element-based approximation to the Lagrange equation of motion.

The deformable model is approximated by a finite number of small regions called elements. The finite elements are assumed to be interconnected at nodal po

ints on their boundaries. The local degree of freedom qd can describe displacements, slope

s and curvatures at selected nodal points on the deformable model.

Introduction The displacement field within the element dj i

s approximated using a finite number of interpolating polynomials called the shape functions

Displacement d anywhere within the deformable model

Introduction Appropriate Elements

Two error components Discretization errors resulting from geometric differences

between the boundaries of the model and its finite element approximation. Can be reduced by using smaller elements

Modeling errors, due to the difference between the true solution and its shape function representation. Shape function errors do not decrease as the element size r

educes and may prevent convergence to the exact solution.

Introduction Appropriate Elements

Two main criteria required of the shape function to guarantee convergence Completeness

Use of polynomials of an appropriate order

Conformity The representations of the variable and its

derivatives must be continuous across inter-element boundaries.

Tessellation

C0 Bilinear Quadrilateral Elements The nodal shape

functions

C0 Bilinear Quadrilateral Elements The derivatives of the sh

ape functions

Integrate a function f(u,v) over Ej by transforming to the reference coordinate system:

North Pole Linear Triangular Elements The nodal shape

functions

Derivatives of the shape functions

North Pole Linear Triangular Elements Integrate a function f

(u,v) over Ej by transforming to the reference coordinate system:

South Pole Linear Triangular Elements The nodal shape

functions

Derivatives of the shape functions

South Pole Linear Triangular Elements Integrate a function f

(u,v) over Ej by transforming to the reference coordinate system:

Mid-Region Triangular Elements

The nodal shape functions

Derivatives of the shape functions

Mid-Region Triangular Elements

Integrate a function f(u,v) over Ej by transforming to the reference coordinate system:

C1 Triangular Elements The relationship between the uv and ξη coor

dinates:

C1 Triangular Elements The nodal shape functions Ni’s

Approximation of the Lagrange Equations Approximation using the finite element

method All quantities necessary for the Lagrange

equations of motion are derived from the same quantities computed independently within each finite element.

Approximation of the Lagrange Equations Quantity that must be integrated over

an element Approximated using the shape functions

and the corresponding nodal quantities.

Example1 When the loads are applied very slowly.

Example1 Consider the complete bar as an

assemblage of 2 two-node bar elements

Assume a linear displacement variation between the nodal points of each element. Linear Shape functions

Example1 Solution.

Black board!!!

Example1 When the external loads are applied

rapidly. Dynamic analysis

No Damping is assumed.

Applied Forces If we know the value of a point force f(u) withi

n an element j, Extrapolate it to the odes of the element using

fi=Ni(u)f(u) Ni is the shape function that corresponds to node i and fi is

the extrapolated value of f(u) to node i.