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Next: Constraints Up: Physically based relaxation of Previous: Problem statement

Mesh relaxation

We can assume that each node in mesh is a mass point and that this mass has connections with springs to other nodes. If string length differs from original, node will be forced into movement. Angles between connection can be similarly constrained.

  figure41
Figure 2: Constraints in mesh: node spacing and angles

Figure 2 shows constraints between nodes. When the length of connection between two nodes is not the desired one, force is applied at each node. Similarly we have torsion strings between connections to force the desired angle. Moment M produced by torsion string is applied at each node as force F= M/(2l), where l is length between two nodes. For each node in mesh governing equation of motion can be written as:
 equation49
Linear spring constant tex2html_wrap_inline316 denotes linear force for shift from stable position tex2html_wrap_inline318. Since stable position is not know at beginning this kind of force must be added to external force tex2html_wrap_inline320. Solution of the homogeneous equation
 equation63
shows that damper constant tex2html_wrap_inline316 should be chosen carefully to achieve fast convergence to stable position. For aperiodic movement the following condition must be satisfied:
 equation71
Note that spring constant has been moved out of equation 2 and only used for dumper constant estimation. Equivalent tex2html_wrap_inline324 needed for equation 4 is obtained from maximum estimated displacement tex2html_wrap_inline326 of node and sum of all external forces tex2html_wrap_inline328 which are connection constraints:
equation81
Equation of motion for each node can be written as:
 equation86

Let tex2html_wrap_inline330 be a distance between two nodes. Virtual movement of both nodes is shown in change of connection length and can be approximated as
 equation96
Change of distance is directly connected with dumper speed it must be applied to both nodes connected. For 3D space the solution of equation 7 is:
 equation102

Equation 6 for motion of single node can be rewritten for all nodes in space state to form system of linear differential equations
  eqnarray119
that can be solved with Euler integration method.




next up previous
Next: Constraints Up: Physically based relaxation of Previous: Problem statement

Leon Kos
Tue Dec 2 10:14:31 CET 1997