1. Physics Based Modeling II Deformable Bodies Lecture 2 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 q d can describe displacements, slopes and curvatures at selected nodal points on the deformable model.

3. Introduction The displacement field within the element d j 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. C 0 Bilinear Quadrilateral Elements The nodal shape functions

8. C 0 Bilinear Quadrilateral Elements The derivatives of the shape functions Integrate a function f(u,v) over E j 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 E j 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 E j 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 E j by transforming to the reference coordinate system:

15. C 1 Triangular Elements The relationship between the uv and ξη coordinates:

16. C 1 Triangular Elements The nodal shape functions N i ’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 f i =N i (u)f(u)  N i is the shape function that corresponds to node i and f i is the extrapolated value of f(u) to node i.