Learn how to improve the efficiency and accuracy of your numerical solutions in mechanical engineering by optimizing the number of iterations for your numerical methods.

Three common criteria to check solution convergence: (1) Check that the residuals have stabilized. (2) Check that the physics laws are satisfied; for example: conservation of mass and conservation of energy. (3) Check that the quantity of interest has reached a stable trend/value.

How do you optimize the number of iterations for a numerical method?

Numerical methods are powerful tools for solving complex problems in mechanical engineering, such as fluid dynamics, heat transfer, stress analysis, and optimization. However, numerical methods often require multiple iterations to converge to an acceptable solution, which can be time-consuming and computationally expensive. How do you optimize the number of iterations for a numerical method? In this article, you will learn some strategies and tips to improve the efficiency and accuracy of your numerical solutions.

1 Define the problem

The first step to optimize the number of iterations for a numerical method is to define the problem clearly and choose the appropriate method for solving it. You should identify the type of equation, the boundary and initial conditions, the domain and discretization, and the expected output and error tolerance. Depending on the problem, you may use different numerical methods, such as finite difference, finite element, finite volume, or spectral methods. Each method has its own advantages and disadvantages, and you should select the one that best suits your problem characteristics and objectives.

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Samuel Lemesa

Aircraft Mechanic at Ethiopian Airlines
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To optimize the number of iterations in a numerical method, use a suitable convergence criteria, choose a good initial guess, implement adaptive methods, use preconditioning techniques, apply problem-specific techniques, and consider parallelization. These strategies can help balance accuracy and computational efficiency.

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Praveen Kumar Loharkar, Ph.D.

Passionate Learner|Educator|Researcher|GenAI EnthusiastlGold Medallist|2x SAS Certified
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Definition of the problem is paramount to achieve the best possible solution. It includes correct definition of boundary conditions and initial values followed by the solution technique/method. We may have multiple ways of solving same problem. Right selection of the numerical methodology can help in optimizing iterations.Although it might seem to be a challenging task initially but a review of releted problems through literature, experience might help to improve the chances of achieving optimum count of iterations.

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2 Choose a solver

The next step is to choose a solver that can handle the numerical method you have chosen. A solver is an algorithm that iterates until it finds a solution that satisfies a certain criterion. There are many types of solvers, such as direct, iterative, linear, nonlinear, explicit, implicit, or hybrid. Some solvers are faster, more stable, or more accurate than others, but they may also have higher memory or computational requirements. You should compare different solvers and evaluate their performance and suitability for your problem.

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3 Set the parameters

The third step is to set the parameters that control the iteration process of the solver. These parameters may include the initial guess, the step size, the relaxation factor, the convergence criterion, and the stopping criterion. The initial guess is the starting point of the iteration, and it should be close to the true solution to reduce the number of iterations. The step size is the increment or decrement of the variable in each iteration, and it should be small enough to ensure accuracy but large enough to avoid stagnation. The relaxation factor is a factor that modifies the step size to improve convergence or stability. The convergence criterion is a measure of how close the solution is to the true value, and it can be based on the absolute or relative error, the residual, or the norm. The stopping criterion is a condition that determines when to stop the iteration, and it can be based on the number of iterations, the convergence criterion, the time limit, or the user input.

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4 Test and validate

The final step is to test and validate the numerical solution obtained by the solver. You should check if the solution satisfies the physical and mathematical constraints of the problem, such as conservation laws, boundary conditions, and symmetry. You should also compare the solution with analytical, experimental, or benchmark results, if available, and quantify the error and uncertainty. You should also analyze the sensitivity and robustness of the solution to changes in the parameters, the method, or the problem. If the solution is not satisfactory, you should go back to the previous steps and modify the problem definition, the method, the solver, or the parameters until you achieve the desired accuracy and efficiency.

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Dr. Mustafa Kaddoura

Innovator | CFD & AI/ML Senior Application Engineer @ ANSYS | Working Towards Sustainable Communities via Digital Engineering
(edited)
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Three common criteria to check solution convergence: (1) Check that the residuals have stabilized. (2) Check that the physics laws are satisfied; for example: conservation of mass and conservation of energy. (3) Check that the quantity of interest has reached a stable trend/value.

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5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Shahabaldin Zangeneh, PhD

Fracture mechanics, failure analysis, and FFS assessment.
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Various solution controls that can be adjusted to optimize the number of iterations. These include parameters related to the solution algorithm, time incrementation, and solution strategies. Experimenting with different solution control settings can help achieve convergence with a reasonable number of iterations.

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Praveen Kumar Loharkar, Ph.D.

Passionate Learner|Educator|Researcher|GenAI EnthusiastlGold Medallist|2x SAS Certified
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Ignoring or avoiding calculation of unwanted parameters might help in optimizing iterations. It might seem very obvious however while defining the problem, one needs to be pretty clear of the objective of the problem. Unnecessary calculations can be avoided thus creating a chance of reducing iterations.

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How do you optimize the number of iterations for a numerical method?

1

2

3

4

5

1 Define the problem

2 Choose a solver

3 Set the parameters

4 Test and validate

5 Here’s what else to consider

Mechanical Engineering

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How do you optimize the number of iterations for a numerical method?

1

2

3

4

5

1 Define the problem

2 Choose a solver

3 Set the parameters

4 Test and validate

5 Here’s what else to consider

Mechanical Engineering

Rate this article

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