Process for computing geometric perturbations for probabilistic analysis
A method for computing geometric perturbations for probabilistic analysis. The probabilistic analysis is based on finite element modeling, in which uncertainties in the modeled system are represented by changes in the nominal geometry of the model, referred to as “perturbations”. These changes are accomplished using displacement vectors, which are computed for each node of a region of interest and are based on mean-value coordinate calculations.
1. A computer-implemented method of computing normalized nodal displacement vectors, which change nodal coordinates of a finite element model in response to a change in a random variable, comprising:
displaying the finite element model;
providing a graphical user interface that receives user input applied to the displayed model in graphical form representing the following data: a region of the finite element model, the region having endpoints, and a displacement direction;
calculating boundary segments and boundary segment endpoint displacements from the input data;
For each segment, using the boundary segment endpoint displacements to compute displacements of border nodes of the segment, thereby providing a set of border node displacement vectors;
Computing mean-value coordinates for each node of the region;
Computing displacement vectors for each node of the region, based on the border node displacement vectors, the input data and the mean-value coordinates;
normalizing the displacement vectors, thereby generating a set of normalized nodal displacement vectors; and
Storing the normalized nodal displacement vectors for use as input to the finite element model.
2. The method of claim 1 , wherein the step of computing displacements of boundary nodes is performed for each segment node by: determining a piecewise distance from a segment endpoint node; normalizing that distance; and interpolating the associated endpoint displacement with the normalized distance.
3. The method of claim 1 , wherein the step of computing mean-value coordinates is performed by using a barycentric calculation method.
4. The method of claim 1 , wherein the step of computing mean-value coordinates is performed for each node, by the following steps applied to each node: accessing node coordinates and associated boundary node coordinates; defining a context based on the foregoing step; computing a homogeneous barycentric coordinates for the node; and using the results of the preceding step to compute mean-value coordinates for the node.
5. The method of claim 1 , wherein the step of computing displacement vectors is performed by the following steps: weighting the magnitude of each border displacement vector by an associated mean value coordinate; and computing an average of the weighted border displacement vectors.
6. A computer-implemented method of changing the nodal coordinates of a finite element model in response to a change in a random variable, comprising:
displaying the finite element model;
providing a graphical user interface that receives user input applied to the model in graphical form representing the following input data: a region of the finite element model, the region having endpoints, and a displacement direction;
calculating boundary segments and boundary segment endpoint displacements from the input data;
for each segment, using the boundary segment endpoint displacements to compute displacements of border nodes of the segment, thereby providing a set of border node displacement vectors;
Computing mean-value coordinates for each node of the region;
Computing displacement vectors for each node of the region, based on the border node displacement vectors, the input data and the mean-value coordinates;
normalizing the displacement vectors, thereby generating a set of normalized nodal displacement vectors;
Storing the normalized nodal displacement vectors for use as input to the finite element model;
Applying a random variable to the normalized nodal displacement vectors; and
Calculating nodal displacements in response to the applying step.
7. The method of claim 6 , further comprising the step of graphically displaying the model with the nodal displacements represented by geometric perturbations of the model.