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Physics Based Modeling II Deformable Bodies
Lecture 2Kwang 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 points on their boundaries. The local degree of freedom qd can describe
displacements, slopes and curvatures at selected nodal points on the deformable model.
Introduction
The displacement field within the element dj is 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 reduces 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 shape 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 coordinates:
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) within 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.