3. CONVERGENCE CRITERIA
• In the context of the Finite Element Method (FEM), convergence criteria refer to the conditions
or rules used to determine when a numerical solution has reached an acceptable level of
accuracy or convergence.
• When solving a problem using FEM, the goal is to find an approximate solution that converges
to the true or exact solution as the computational mesh is refined.
• Convergence criteria are essential to determine when to stop refining the mesh or iterating the
solution process, as excessive refinement can be computationally expensive and unnecessary.
• As we go on increasing the number of elements the accuracy of the solution goes on improving
(i.e the solution moves more closer to the exact solution)
12-10-2023 WRHI, GEC TSR 3
5. CONVERGENCE CRITERIA (contd…)
• Common convergence criteria in FEM include:
Residual Norm: The residual is the difference between the calculated solution and the true solution.
Convergence can be achieved when the norm of the residual falls below a specified tolerance. The
residual norm is often used for iterative methods like the conjugate gradient method or the Newton-
Raphson method.
Relative Error: Convergence can also be assessed by comparing the current solution to the previous
one. If the relative change in the solution falls below a predetermined tolerance, convergence is
achieved.
12-10-2023 WRHI, GEC TSR 5
6. CONVERGENCE CRITERIA (contd…)
Convergence Rate: Examining the rate at which the solution approaches a steady state can also be a
convergence criterion. If the rate decreases and stabilizes within acceptable limits, the solution can be
considered converged.
Grid Independence: This criterion involves performing simulations on progressively finer meshes
(grids) and observing how the results change. Convergence is typically achieved when the results
become grid-independent, indicating that further mesh refinement is unnecessary.
Energy Norm: In some cases, the energy norm of the error can be used as a convergence criterion. It
quantifies the error in terms of the underlying physics and can be more relevant for certain problems.
12-10-2023 WRHI, GEC TSR 6
7. CONVERGENCE CRITERIA (contd…)
• The choice of convergence criteria depends on the specific problem being solved and the numerical
techniques employed.
• It's important to strike a balance between achieving sufficient accuracy and minimizing
computational costs.
• Engineers and scientists often perform convergence studies to determine appropriate convergence
criteria and ensure the reliability of their FEM simulations.
12-10-2023 WRHI, GEC TSR 7
8. CONVERGENCE REQUIREMENTS
1. The field variable must be continuous within the elements. This requirement is easily satisfied by
choosing continuous functions as interpolation models. Since polynomials are inherently
continuous, the polynomial type of interpolation models satisfy the requirements.
2. The interpolation polynomial must be able to give a constant value of the field variable within the
element when the nodal values are numerically identical.
3. The field variable Φ and its partial derivatives upto one order less than the highest order derivative
appearing in the functional I(Φ) must be continuous at element boundaries or interfaces.
12-10-2023 WRHI, GEC TSR 8
9. CONFORMING AND NON CONFORMING
ELEMENTS
• In the context of the Finite Element Method (FEM) and convergence criteria, the terms
"conforming" and "non-conforming" refer to how elements (subdivisions of the computational
domain) are connected at their common interfaces.
• These concepts are essential for understanding how elements interact and how convergence
criteria are applied in FEM simulations.
12-10-2023 WRHI, GEC TSR 9
10. CONFORMING AND NON CONFORMING
ELEMENTS
Conforming Elements
• Conforming elements are also known as "continuous" or "compatible" elements.
• In conforming FEM, elements are constructed in such a way that they share common nodes or
vertices along their boundaries.
• This ensures that the variables being approximated (e.g., displacements, temperatures) are continuous
across element boundaries.
• Conforming elements are typically used in problems where smooth solutions are expected or
required, such as problems involving heat transfer, elasticity, or fluid flow.
• Conforming elements simplify the application of convergence criteria because the solution within the
elements and at their interfaces can be easily interpolated and integrated.
12-10-2023 WRHI, GEC TSR 10
11. CONFORMING AND NON CONFORMING
ELEMENTS
•
Non Conforming Elements
• Also known as "discontinuous" or "incompatible" elements.
• In non-conforming FEM, elements are constructed in a way that they do not necessarily
share common nodes or vertices along their boundaries.
• This can result in discontinuities in the approximated variables at element interfaces.
12-10-2023 WRHI, GEC TSR 11
12. CONFORMING AND NON CONFORMING
ELEMENTS •
Non Conforming Elements
• Non-conforming elements are often used in problems where discontinuities or singularities
in the solution are expected, such as problems involving crack propagation, contact
mechanics, or domain decomposition techniques.
• Convergence criteria for non-conforming FEM can be more challenging to apply because of
the discontinuities at element interfaces. Special techniques and criteria may be needed to
assess convergence in these cases.
12-10-2023 WRHI, GEC TSR 12
13. CONFORMING AND NON CONFORMING
ELEMENTS •
• When it comes to convergence criteria, the choice between conforming and non-conforming
elements can impact how the criteria are applied and interpreted.
• For conforming elements, convergence criteria typically focus on assessing the accuracy of the
solution within each element and ensuring that it approaches the true solution as the mesh is refined.
The continuity of variables at element interfaces simplifies this process.
• For non-conforming elements, convergence criteria may involve evaluating the convergence of
interface conditions or the matching of values at discontinuities. Techniques such as jump indicators
or error estimators that consider the discontinuities are often used to assess convergence.
12-10-2023 WRHI, GEC TSR 13
14. CONFORMING AND NON CONFORMING
ELEMENTS •
• The choice between conforming and non-conforming elements depends on the nature of the problem
being solved and the desired level of accuracy and realism in the solution.
• Conforming elements are generally more straightforward to work with in terms of convergence
assessment, but non-conforming elements are necessary for problems with specific geometric or
physical characteristics
12-10-2023 WRHI, GEC TSR 14
15. REFERENCES
• Finite Element Analysis: Theory and Application with ANSYS" by
Saeed Moaveni.
• "Introduction to the Finite Element Method" by J.N. Reddy.
• "The Finite Element Method: Its Basis and Fundamentals" by O.C.
Zienkiewicz and R.L. Taylor.
12-10-2023 WRHI, GEC TSR 15
