P-Norm
Add a P-norm node () by right-clicking a Definitions node in a Component and choosing it from the Physics Utilities submenu. It is also available on the Definitions ribbon toolbar, from the Physics Utilities menu.
The p-norm of an expression over a selection (domain, edge, boundary, or point) is a differentiable approximation of its maximum absolute value on the selection. An exact maximum is not differentiable and therefore not compatible with gradient-based optimization.
P-Norm
Enter the Expression whose maximum value is sought. The approximate maximum is made available as a global variable with the name specified in the Name field. Choose the exponent P in the approximation directly from the Norm list or select User defined to set an expression for the exponent. A larger norm value improves the approximation. In practice, however, norm values that are excessively large can cause problems for the solvers. When this happens, you can try to increase the exponent step by step using a parameter-dependent User defined expression.
Quadrature Settings
The default Quadrature method is Gauss, which uses a pattern of integration points in the interior of mesh elements. Choose instead Gauss-Lobatto to use a different pattern including points at mesh element nodes. These predefined patterns are constructed so that a polynomial of at most the specified Integration order will be integrated exactly. If you have added custom Integration Rule nodes under Global Definitions, these are also available in the Quadrature method setting.
For the Gauss and Gauss-Lobatto quadrature methods, choose a suggested Integration order or User defined which will expose the Order expression field. The suggested even orders from 0 to 8 are available in all dimensions. The default is 4 which is appropriate for a second-order equation using quadratic shape functions. An order entered as a user-defined parameter expression must evaluate to an integer, but may be odd. The maximum order depends on the space dimension. It is 30 in 2D and 14 in 3D.
Choose a frame from the Integrate on frame list to set the volume element in the weak form integral. The default is Spatial.
The P-norm is symmetric in the expression, but the influence of negative values can be avoided using a soft max function in the expression, that is, .
Figure 5-1: A discrete p-norm with two variables is illustrated for p = 5 and p = 20 showing how this approximation to max(x,y) improves with higher values of p.
Optimizing a Flywheel Profile: Application Library path Optimization_Module/Shape_Optimization/flywheel_profile