The uniform model has a starting mesh of 23,954 elements refined to 64,627 elements. This required the longest time to solve, with about 6% error. It is the least accurate of the methods.

The uniform model has a starting mesh of 23,954 elements refined to 64,627 elements. This required the longest time to solve, with about 6% error. It is the least accurate of the methods.

The h-method resolved at 15,492 elements with a faster response and accuracy within 1%.

The h-method resolved at 15,492 elements with a faster response and accuracy within 1%.

The p-method resolved at 3,835 elements with the fastest solution time and within 3% accuracy.

The p-method resolved at 3,835 elements with the fastest solution time and within 3% accuracy.

The hp method with 3,958 elements is slightly slower than the p-case but accuracy is within 1%.

The hp method with 3,958 elements is slightly slower than the p-case but accuracy is within 1%.


The most widely used methods are the h and p-adaptive refinement. Each has its trade-offs. But combining the two into an advanced form improves accuracy and processing time while eliminating many of the trade-offs.

"In an ideal world, the accuracy of FEA results is determined by the user knowing the exact solution for a particular mode," says Tomi Mossessian, CEO, PlassoTech Inc., Encino, Calif. "But the solution is not available in most cases, so a process has to be established for reaching a specified accuracy. The automated procedure for improving results that reach the required level of accuracy is called an adaptive process," he says.

The most common way to check the effectiveness of an adaptive approach is the rate of convergence. "It shows how fast the solver is approaching an exact solution as the user increases the discretization degrees of freedom," says Mossessian. The compute efficiency of the method is also an important factor because users want to expend the least amount of time and memory to get the most accurate results.

Mossessian has experimented with several methods to show the pros and cons of each. For example, the uniform mesh refinement method is one of the simplest for improving FEA results. "It creates a finer mesh by uniformly decreasing the element size throughout the model. This increases the number of elements until the results stop substantially changing. That is, the solution has converged within a certain percentage," he says.

Although the simplest method, it is also the least efficient because a uniformly finer mesh of a 3D model substantially increases the number of elements to process. "For example, a 1-m cube containing cubic elements of 10 cm in lengths would have 1,000 elements, by reducing the element size to 5 cm, the number of elements increases to 8,000. The growth is cubic," says Mossessian. Even in thin solids the growth is quadratic. "The uniform mesh refinement method is too slow and resource intensive to be practical in most real-life design situations," he says.

An h-adaptive method improves on uniform mesh refinement. H-adaptive refines the mesh only in areas containing a high number of errors. The increase in the number of elements that must be processed is substantially less than when uniformly refining the mesh. Technical papers often refer to these methods as h-adaptive.

The technique typically uses an error estimate over the elements at a given solution step and refines the mesh by a practical factor only in the high-error regions and then resolves the problem. The adaptive mesh refinement and resolving continues until reaching the specified accuracy. For structural analysis, typically this would be highest stress in the model. Although the h-adaptive method is more efficient than uniform refinement, for many cases it does not have an optimum convergence rate and requires complete or partial remeshing of the model in addition to resolving at each cycle, which is a time-consuming process," says Mossessian.

The p-adaptive method does not change the mesh or number of elements. Instead, it increases the order of the polynomial approximation used within each element. This eliminates the need for remeshing and only requires resolving the problem at each p-cycle until the required accuracy is reached.

"Similar to the h-adaptive method, error estimates are used with different points of the mesh. Polynomial order is increased within regions of high error until results reach the user-defined accuracy. In terms of convergence rate, this approach is superior to the traditional h-adaptive method. However, when sharp stress concentrations are present, this advantage deteriorates. In addition, by substantially increasing the p-order, it becomes compute expensive," he says.

The best method for efficiently reaching convergence is a combination of both methods, the hp-adaptive method. It provides the most-efficient technique for controlling the FEA approximation errors. Rather than increasing the p order indefinitely or just relying on pure mesh refinement through the h-adaptive method, the hpmethod uses the p-adaptive within each h-refinement step. This is the most difficult method to implement and is offered by few FEA packages.

The implementation of FEA solvers based on the hp-adaptive method in 3G.author, an FEA design-analysis program, allows calculating accurate results in the most-efficient manner. This allows a versatile definition of convergence criteria for both global and local parameters and provides a number of advanced attributes with respect to accuracy, efficiency, and robustness.

A key attribute is an efficient p-based solver using a state of the PCCG ( preconditioned conjugate gradient) technique tied to adaptive mesh refinement. In addition, accuracy can be controlled so it is easily tailored for global or local result parameters, including stresses, displacement and temperature. Localized results can include or exclude any geometric entity, such as faces, edges, and vertices.

In addition, advanced functions allow controlling accuracy and efficiency when solving assembly-type models.

For users needing complete control over meshing, Mossessian says, it is available on top of the hp-adaptive method. "Users more familiar with their model behavior can accelerate the adaptive process by choosing a finer initial mesh over any area of interest, such as parts, faces, edges and vertices, using 3G.author's global and local mesh control capabilities.

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