Physics Based Modeling II Deformable Bodies

1. Physics Based Modeling IIDeformable Bodies Lecture 12 Kwang Hee Ko Gwangju Institute of Science and Technology

2. 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.

3. 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

4. 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.

5. 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.

6. Tessellation

7. C0 Bilinear Quadrilateral Elements • The nodal shape functions

8. 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:

9. North Pole Linear Triangular Elements • The nodal shape functions • Derivatives of the shape functions

10. North Pole Linear Triangular Elements • Integrate a function f(u,v) over Ej by transforming to the reference coordinate system:

11. South Pole Linear Triangular Elements • The nodal shape functions • Derivatives of the shape functions

12. South Pole Linear Triangular Elements • Integrate a function f(u,v) over Ej by transforming to the reference coordinate system:

13. Mid-Region Triangular Elements • The nodal shape functions • Derivatives of the shape functions

14. Mid-Region Triangular Elements • Integrate a function f(u,v) over Ej by transforming to the reference coordinate system:

15. C1 Triangular Elements • The relationship between the uv and ξηcoordinates:

16. C1 Triangular Elements • The nodal shape functions Ni’s

17. 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.

18. Approximation of the Lagrange Equations • Quantity that must be integrated over an element • Approximated using the shape functions and the corresponding nodal quantities.

19. Example1 • When the loads are applied very slowly.

20. 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

21. Example1 • Solution. • Black board!!!

22. Example1 • When the external loads are applied rapidly. • Dynamic analysis • No Damping is assumed.

23. 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.