Finite element analysis (FEA) has significantly transformed the approach of engineers and scientists to design and analyze various physical systems. It has emerged as a formidable tool for enhancing designs and detecting possible problems by providing valuable insights into system behavior without the need for costly and time-consuming physical prototypes. However, while simulating physical system behavior using FEA, it is very important to validate and verify the results to ensure the accuracy and reliability of them.

We often hear about convergence issues, which can arise from multiple underlying sources while working with FEA problems. Convergence is a crucial aspect of FEA, which refers to the state of the computed solution – where it becomes stable and does not change significantly with further refinement of the numerical model parameters such as mesh density or iteration steps. In this article, we will learn about different types of convergence and how to tackle them to attain a reliable numerical solution.

## Mesh Convergence

FEA is a computational method used to solve the equation of motion of a physical system describing its behavior. In this numerical technique, the physical domain is divided into smaller, finite sized, elements to calculate approximate behavior. The solution coming from FEA is an approximate solution, which is highly dependent on mesh size and the type of elements, which can often lead to mesh convergence issues. There are two methods to overcome the mesh convergence issues, and these are discussed below.

### H-Based Method

In H-based methods, the physical system is meshed using simple first order linear or quadratic elements. The accuracy of the solution is improved by increasing the number of elements in the model. The computational time increases with increase in number of elements. This method is used in Abaqus predominantly. With increase in the refinement of the elements, the solution gets closer and closer to the analytical solution. The goal here is to find the mesh resolution where further refinement is not going to significantly alter the results. This mesh refinement is not applicable in some situations leading to singular solutions like reentrant corners, crack tips in fracture analyses, concentrated forces applied to single nodes and localized material damage problems.

The H method is demonstrated using a simple FEA model below. As we can see that, with decrease in element size, the nominal stress in the model is reaching a converged value of 13.6 MPa.

### P Based Method

In this method, the number of elements is kept minimal, and the convergence is achieved by increasing the order of the elements (4th , 5th or 6th). The computational time increases again with increasing the order of elements as the number of degrees of freedom increases exponentially with the element order. The figure below demonstrates the P method. We can see that the nominal stress quickly reaches its asymptotic value by changing the order of the element.

Achieving mesh convergence is very important as it ensures that the obtained FEA solution is not highly dependent on the mesh size and that it accurately represents the physical behavior of the system. It increases the confidence inaccuracy of the numerical results and helps engineers to make sound decisions about the design and analysis of the structures. This can be computationally very expensive as it requires multiple iterations. So, it is essential for the engineer to use their judgement and company-based guidelines to attain a balance between mesh refinement and computational resources.

## Time Integration Accuracy

This is a very important aspect of finite element analysis, especially in dynamic simulations that are highly time-dependent, such as structural vibrations, impact analysis and transient thermal behavior. In these dynamic simulations, the equation of motion must be integrated over time. So, the size of the time step must be small enough to capture any phenomenon occurring in the analysis and to achieve reliable and meaningful results in dynamic simulations.

Abaqus provides user specified parameters to control time integration accuracy like half-increment residual tolerance, maximum temperature change allowed in an increment and maximum difference in creep strain allowed in an increment. Using these parameters, we can ensure that all the nonlinear behavior is captured during the analysis.

Higher-order time integration methods, such as implicit and explicit Runge-Kutta methods can be used in simulations where higher accuracy is required. But these higher accuracy methods are associated with higher computational costs.

## Convergence Of Nonlinear Solution

When the external loads (**P**) are applied on a system, it is subjected to internal forces (**I**) due to stresses generated in the system. It is said to be in equilibrium when the external forces are balanced by internal forces.

**P â€“ I = 0**

For linear problems, this equation will have a unique solution. The solution **u** is calculated by only using the current value of **P.**

**P â€“ Ku =0**

But when nonlinearity is introduced into the FEA model through material, boundary (contact, friction) and geometry (large deflections/rotations), the stiffness of the structure changes as it deforms under the applied loads. For more detailed information regarding nonlinearity, please refer here. An example force- displacement curve for a nonlinear model is shown below.

The above equilibrium equation will not have a unique solution when nonlinearity is present in the FEA model. The equation may have zero, one, many or infinite number of solutions.

The solution **u** depends on the entire load history of **P**. To solve these nonlinear problems, a common technique is to break the total load applied into small incremental loads. With increase in nonlinearity in load application, the increment size reduces, leading to an increase in computational costs. An approximate solution is calculated for each load increment using several iterations. The two robust iterative methods used for solving nonlinear problems are Newton-Raphson and Quasi-Newton techniques. In the Newton-Raphson method, the stiffness matrix is calculated in every iteration, whereas it is not recalculated for every iteration in Quasi-Newton method. A sufficiently accurate solution is calculated for each iteration using these methods by specifying tolerances.

**P â€“ I = R â‰¤ Tolerances**

## Solution Convergence

A converged solution refers to a stable and consistent solution as the analysis progresses. To attain this, all the above discussed sources of convergence errors need to be tackled: Mesh, time and load integration accuracy and nonlinearity in the model. Further, this also requires good engineering judgement in creating analysis models with proper materials, loads, boundary conditions and solution procedures. The goal here is to achieve a solution where the parameters, such as residuals and errors, decrease to a minimal level and remain constant.

## Final Thoughts

Convergence criteria is highly significant in FEA as it determines the accuracy and reliability of the obtained numerical results. The computed solution may not produce the actual behavior of the physical system if convergence is not reached. Hence, analysts have to perform sensitivity studies by changing parameters like mesh density, time step and load increments to be sure that they have reached a converged solution.

Hopefully, this article has given a good overview of sources for convergence and how to tackle them. Weâ€™re always here to help, so if you have questions about convergence in your models, or just FEA in general, ,**donâ€™t hesitate to reach out**!