Convergence with Mesh Refinement

Expected completion time: 15 minutes.

Use with: Inventor 2010 and later

When you set up your simulation, you modify or accept the default settings for the convergence criteria. The Stress Analysis solver uses these conditions to determine when to stop the solution. The results of this iterative refinement process are displayed on the convergence plot.

In this Skill Builder, you are introduced to convergence as it relates to Inventor Stress Analysis. Images and animations demonstrate the following topics:

• Convergence and results
• Convergence settings
• Convergence plots

Overview

To perform a Finite Element Analysis (FEA) in Inventor Stress Analysis, the physical structure is discretized into a mesh of finite elements. A corresponding set of equations, based on Elasticity theory, is solved during the analysis to determine the behavior of the structure under the assigned boundary conditions.

The discrete FE model contains approximation errors, given that the actual structure is a continuum. Appropriate mesh refinements can help reduce these errors and result in better approximations of the mathematical model. The iterative sequence in which the FEA results are evaluated, the mesh redefined, and the equation set solved embodies the convergence process.

Inventor Stress Analysis efficient solver generates fast and accurate results by performing (h-p) adaptive mesh refinements throughout the solution iterations. Adaptive h refinement changes the size of elements locally and usually produces a finer mesh, while adaptive p refinement escalates the polynomial order of selected elements. The (h-p) adaptive refinement is a cyclic procedure. It retrieves information from intermediate solutions and subsequently adjusts the mesh locally to improve accuracy of results. This refinement progression is fully automated in Inventor Stress Analysis.

Convergence and Results

Initially, a termination convergence tolerance (for example, results error less than10%) is set and the iterative solving process started. The problem solution is said to converge, and the iterative cycles stop when this condition is met. This criterion can be used as the likely error measure.

On the other hand, if the difference between intermediate results increases without bound, the solution is said to diverge. The iterative cycle stops after reaching the assigned number of refinements. Divergence usually occurs at areas of sharp geometry involving concave angles and can also occur at boundary conditions depending on the FE model definition. Stress singularities, or locations of theoretical infinite stress due to point loads and constraints, also cause divergence.

When evaluating your solution results, it is important that the FE model behavior agrees with your expectations of the physical part. Check for proper constraints, loads, contact conditions, material properties, and so on. Modeling mistakes in these areas usually lead to incorrect results regardless of convergence. A solution can still converge, although to a wrong result, if the process starts with an incorrect FE model.

The solution of FE models involves extensive computation which typically introduces small errors due to truncation and numerical round-off. Inventor Stress Analysis uses advanced numerical analysis methods to limit numerical round-off and ill-conditioning. However, some FE models can amplify these small errors with significant effects.

Convergence Settings

The following image shows the Convergence Settings dialog box for a static analysis. In this section ,we input the settings by which the solver determines if convergence is met.

Maximum Number of h Refinements specifies the maximum number of mesh refinement cycles. If the solution converges before reaching this maximum number, the h refinement iterations stop.

Stop Criteria (%) specifies the convergence goal. If the solutions for the selected results, between two consecutive iterations, differ by less than the specified percentage, the refinement process stops and the last results display.

Convergence Plots

When the Maximum number of h refinements is greater than zero, you can create an XY plot of the convergence. For example, the following image shows a typical convergence trend for parts and assemblies. In this example, the Von Mises Stress was specified as the Results to Converge and is plotted versus the Solution Step. The overall convergence rate is shown at the top of the plot.

For Part analysis, the solution steps 1, 2 and 3 represent the polynomial escalations p1, p2 and p3, respectively. The solution steps 4, 5, …,n correspond to the combined h-p refinement results for h = 1, 2, …,(n-3), respectively.

For Assembly analysis, the solution steps 1 and 2 represent the polynomial escalations p1 and p2, respectively. The solution steps 3, 4, ...,n correspond to the combined h-p refinements results for h = 1, 2, ...,(n-2), respectively.

Convergence Animations

The following animations show how to use convergence in Inventor Stress Analysis.

Summary

Appropriate use of convergence settings and convergence plots can help improve the accuracy of Stress Analysis results. Always compare results to expectations and make FE model changes as needed.

There are various types of errors that can affect the FEA solution:

• Discretization errors because a FE model has a finite number of elements, while the actual physical model is a continua.
• FE modeling mistakes in representing the behavior of the actual physical model. These errors lead to incorrect results regardless of convergence behavior.
• Numerical errors inherited from computations. These errors are typically small, although some FE models can amplify them.

Mesh refinements enhance the FE model to better approximate the mathematical representation and obtain more accurate results. The Inventor Stress Analysis automatic (h-p) adaptive mesh refinement technology can help you generate fast and accurate solutions with minimum interaction.