Smoothing Methods
In the domains with free displacement, the Moving Mesh interface solves an equation for the mesh displacement. This equation smoothly deforms the mesh given the constraints placed on the boundaries. Choose between Laplace smoothing, Winslow smoothing, hyperelastic smoothing, and Yeoh smoothing.
To specify the smoothing methods, use the Mesh smoothing type list in the Free Deformation Settings section of the Moving Mesh or Deformed Geometry node. To see how these smoothing methods differ, let x and y be the spatial coordinates of the spatial frame, and let X and Y be the reference coordinates of the material frame.
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If Laplace smoothing is selected, the software introduces deformed mesh positions x and y as degrees of freedom in the model. In the static case, it solves the equation
and in the transient case, it solves the equation
Similar equations hold for the y coordinate.
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If Winslow smoothing is selected, the software solves the equation
and does the same for Y. Equivalently, X and Y satisfy Laplace equations as functions of the x and y coordinates.
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The hyperelastic smoothing method searches for a minimum of a mesh deformation energy inspired by neo-Hookean materials:
where μ and κ are artificial shear and bulk moduli, respectively, and the invariants J and I1 are given by
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The Yeoh smoothing method is also inspired by hyperelastic materials, in this case the three-term Yeoh hyperelastic model, which is a generalization of a neo-Hookean material. It uses a strain energy of the form
where κ is an artificial bulk modulus, as above, while C1, C2, and C3 are other artificial material properties. The values of C1 and C3 are by default 1 and 0, respectively, and can only be changed in the Equation View subnodes under a Free Deformation node. The value of C2 controls the nonlinear stiffening of the artificial material under deformation. It is specified in the Stiffening factor field, with a default value of 100.
The Laplace smoothing is the cheapest option in terms of computations since it is linear and uses one equation for each coordinate direction, which are not coupled to each other. On the other hand, there is no mechanism in Laplace smoothing that prevents inversion of elements. Therefore, the method is most suitable for small deformations in a linear regime — for example, when computing the sensitivity of some quantity to small deformations around the initial shape.
The Winslow, hyperelastic, and Yeoh smoothing methods are increasingly nonlinear and create a single coupled system of equations for all coordinate directions, which makes them more expensive to solve. They also share the theoretical property that continuous solutions to these equations always have positive volume everywhere. Unfortunately, this is not necessarily true for the discrete finite element solutions. In addition, a positive volume is not sufficient for maintaining element quality.
In compression, the three nonlinear methods show similar behavior, while in extension, the Winslow smoothing tends to allow elements to be stretched too far. The main difference between the simpler Hyperelastic method and the more advanced Yeoh model is that the latter responds to element distortion by sharply increasing the stiffness of distorted elements. This to some extent prevents further distortion in those regions and effectively acts to spread the mesh deformation more evenly over the domain, away from moving boundaries.
Yeoh smoothing generally produces the best results and allows the largest displacement of boundaries before mesh elements become inverted. However, because of its strong nonlinearity, it can cause convergence problems, in particular for the time-dependent and segregated solvers.