1. 1 THE ACCURACY OF FEA

2. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled in the convergence process

3. 3 DISCRETIZATION OF STRESS DISTRIBUTION Mesh built with first order triangular elements called constant stress triangles First order element assumes linear distribution of displacements within each element. Strain, being derivative of displacement, is constant within each element. Stress is also constant because it is calculated based on strain. Discrete stress distribution in constant stress triangles

4. 4 Tensile hollow strip modelled with a coarse mesh of 2D plate elements. An isometric view of von Mises effective stress distribution in the upper right quarter of the model shown above. The height of bars represents the magnitude of stress. Notice that stresses are constant within each element. DISCRETIZATION OF STRESS DISTRIBUTION

5. 5 CONVERGENCE ANALYSIS BY MESH REFINEMENT The same tensile strip modelled three times with increasingly refined meshes. The process of pprogressive mesh refinement is called h convergence

6. 6 CHARACTERISTIC ELEMENT SIZE The process of pprogressive mesh refinement is called h convergence because characteristic element size h is modified during this process

7. 7 Discretization errors Discretization error is an inherent part of FEA. It is the price we pay for discretization of a continuous structure. Discretization error can be defined either as solution error or convergence error. Convergence error Convergence error is the difference between two consecutive mesh refinements and/or element order upgrade. Let’s say convergence error is 10%. If convergence takes place, then the next refinement and/or element order upgrade will produce results that will be different from the current one by less than 10%. Solution error The solution error is the difference between the results produced by a discrete model with a finite number of elements and the results that would be produced by a hypothetical model with an infinite number of infinitesimal elements. To estimate the solution error, one has to assess the rate of convergence and predict changes in results within the next few iterations as if they were performed. CONVERGENCE CURVE 123 MESH REFINEMENT AND / OR ELEMENT ORDER UPGRADE NUMBER CONVERGENCE CRITERION SOLUTION OF THE HYPOTHETICAL “INFINITE” FINITE ELEMENT MODEL (UNKNOWN) SOLUTION ERROR FOR MODEL # 3 CONVERGENCE ERROR FOR MODEL # 3 # OF D.O.F. IN THE MODEL

8. 8 CONVERGENCE PROCESS WITH h ELEMENTS AND p ELEMENTS The name h comes from characteristic element size usually denoted as h. That characteristic element size is reduced during h convergence process. The name p comes from polynomial function describing displacement field in the element. The order of polynomial function is increased during p convergence process. Both in h and p elements convergence process means adding degrees of freedom the the model. With h elements this is accomplished by mesh refinement. With p elements degrees of freedom and added by increasing elements order while mesh remains unchanged. h - elementsp - elements

9. 9 h - elementsp - elements Element shape: tetrahedral, wedge, hexahedral Mapping allows for only little deviation from the ideal shape. Displacement field mapped by lower order polynomials (1 st or 2 nd ), polynomial order does not change during solution Mapping allows for higher deviation from the ideal shape but may introduce errors on highly curved edges and surfaces Displacement field described by mapped higher order polynomials (up to 9 th ), polynomial order adjusted automatically to meet user’s accuracy requirements. COMPARISON BETWEEN h ELEMENTS AND p ELEMENTS results are produced in the iterative process that continues until the known, user specified accuracy, has been obtained results are produced in one single run with unknown accuracy fewer large elements typically 500 - 10000 many small elements typically 5000 - 500000 Note: Only tetrahedral elements elements can be reliably created with the available automeshers

10. 10 ec001 HOLLOW TENSILE STRIP model fileec001 model typesolid materialalloy steel restraintsbuilt-in to left edge load100,000N tensile load to right edge in x direction objectives meshing solid CAD geometry using solid elements demonstrating h convergence process 100,000 N tensile load Built-in support

11. 11 ec001HOLLOW TENSILE STRIP Step 1 of h convergence process Maximum von Mises stress 345.3 MPa Second order (high quality) solid elements used

12. 12 ec001HOLLOW TENSILE STRIP Second order (high quality) solid elements used Step 2 of h convergence process Maximum von Mises stress 367.7 MPa

13. 13 Step 3 of h convergence process Maximum von Mises stress 375.3 MPa ec001HOLLOW TENSILE STRIP Second order (high quality) solid elements used

14. 14 ARTIFICIAL STIFNESS ELEMENT STRESSES NODAL STRESSES H – ADAPTIVE SOLUTION

15. 15 von Mises stress [MPa] ec001 HOLLOW TENSILE STRIP 345 MPa368 MPa375 MPa Results of h convergence process presented in the form of a convergence curve Verification of stress results: Haywood formula, Peterson p. 111 Ratio of global element size at iteration # 1 to current global element size