Mesh Convergence Study Using ANSYS

Abstract: A finite element analysis for stress and displacement analysis iscommonly used in the mechanical industry. The accuracy of the analysis resultsdepends on the number of elements used for the FE analysis. The FE solutionapproaches to analytical (Exact) solution when number of element (Nodes)increases in the model. The accuracy of the finite element analysis solution for agiven problem and mesh density can be measured in terms of descretizationerror. In this paper different methods to find out the mesh descretization error arediscussed. The mesh convergence is studied for different element types and byvarying the mesh density. ANSYS commercially available analysis package wasused for the analysis.

Introduction: Displacement results are the primary results of a finite elementanalysis and other results are derived from the displacements. Displacementresults are less sensitive to the number of elements or nodes in the model, butstress results vary much with increasing mesh density till the convergence isachieved. So it is very important to validate the stress results before concludingthe analysis.A finite element analysis requires the idealization of an actual physical problem into a mathematical model and then the finite element solution of that model. Thesolution should converge (when the number of elements increased in the model)to the analytical (Exact) solution of the differential equations that govern theresponse of the mathematical model [1]. Finite element solution approaches toanalytical (Exact) solution when number of elements (Nodes) increases in themodel.In linear elastic analysis there is a unique exact solution to the mathematicalmodel (i.e. for given stress analysis problem). It is also important to note thatconvergence is directly related to the load and constraints applied to specific run.Hence the mesh which is converged for particular loading and constraints maynot converge to different loads and constraints.

Nivrutti Garud,Engineer,Satyam Engineering Services Ltd.Secunderabad. E-mail: nivrutti_garud@satyam.com

Elements must be complete and the element and mesh must be compatible toinsure the convergence for a given finite element mesh. The requirement ofcompleteness of an element means that the displacement functions of theelement must be able to represent the rigid body displacements and the constantstrain rates [1].The requirement of compatibility means that the displacements within theelement and across the element boundaries must be continuous. Physicallycomparability insures that no gaps occur between elements when theassemblage is loaded [1].

Error Estimation:For mesh convergence to the exact results the elements must be complete andcompatible. Using compatible elements mean that in the finite element problemsthe displacements and their derivatives are continuous across elementboundaries. And the elemental stresses are calculated using derivatives of thedisplacements and must be continuous across the element boundaries. But thestresses obtained at an element edge (or face), when calculated in adjacentelements may differ substantially if a course finite element mesh is used [1]. Thisstress difference across element boundary decrease as the finite element meshis refined.The stress jumps or stress difference across the element boundaries of the bodyare of course a consequence of the fact that stress equilibrium is not accuratelysatisfied unless a very fine mesh is used. Thus this stress jumps or stressgradient across element boundaries can be used as the measure ofdescretization error for a given mesh [1,2,3].

Methods of Error estimation:1) Using Elemental Stresses:The discontinuity of stress across the element boundaries can be used for theerror estimation. In order to establish a measure of stress difference across theelements it can be compared with the absolute maximum stress value thatoccurs anywhere in the model [3].The error can be estimated using following formula [2].

Error = ((Si(max)-SJ(min))/Smodelmax)*100Where

Si(max) -is maximum elemental stress at element I S(min) -is minimum elemental stress at element j (Adjacent element to element i) Smodelmax -is maximum elemental stress in the model.

Following example gives the error estimation for given four elements. Element i isthe element for which the stress is maximum and element j is element for whichstress is minimum.

Figure 1: Element stress plot.

2) Using Nodal stresses:Similarly nodal stress can be used to find out the error in mesh convergence.

Error = ((Siu-Sia)/Smodelmax)*100

Siu -is maximum unaveraged stress (Elemental stress) at node i. Sia -is average nodal stress at node i. Smodelmax -is maximum averaged stress in the model.

Analysis and results:General plane stress problem (Plate with hole) was considered for this study.The finite element model was modeled using quarter symmetry. Plane stress fournoded element (Plane 42), ten noded tetrahedron element (Solid 92) and eightnoded hexahedral element (Solid 45) were used to mesh the model and separateerror estimation study was carried out. The element density was increaseduniformly throughout the volume to study the effect of descretization. ANSYSV5.6.2 is used as pre and post processor. Steel material properties, YoungsModulus 2.1e5 MPa and Poisons ratio 0.3 was used in the analysis.

Element i

Element j

Error = ((Si-Sj)/Si)*100

Stress in X-direction (longitudinal stress) Sx is used for error estimation. Theplate dimensions (in mm) are shown in Figure 2 below. Thickness of plate istaken as 10mm. Symmetric boundary conditions were applied at cut boundariesand force of 10000N was applied over the end as shown in the Figure 2.

Figure 2: Geometry and Boundary Conditions for the analysis.

Observation 1:The averaged nodal stress (using PLNSOL command in ANSYS) andunaveraged nodal stress (using PLESOL command in ANSYS) contours aregiven in Figure 3 and Figure 4 respectively. It can be seen that the maximumaveraged nodal stress (using PLNSOL command) and maximum unaveragedstress (using PLESOL command) are not at the same node location. Thesestress values are at different node locations and hence can not be used directlyfor error estimation. In the present work the averaged and unaveraged stress atthe same node location was considered for the error estimation.

Symmetric Boundary conditions on these faces.

10000N onthis face.

80

40 20

Figure 3. Stress plot-using PLNSOL.

Figure 4: Stress plot using PLESOL.

Observation 2:If the maximum stress occurs at a corner node, and this node belongs to onlyone element, then the averaged and unaveraged stress at that node will remainsame. In this case the error will be zero percent even if the mesh is course mesh.Following Figure 5 and Figure 6 gives the stress plots for problem wheremaximum stress occurs at the corner node belonging to one element.

Figure 5: Stress plot-using PLNSOL.

Figure 6: Stress plot-using PLESOL.

Also the averaging is generally done considering only the selected elementswhile plotting the stress contour. In this case actual averaged stress consideringall the elements attached to that node may vary with the averaged stress whenonly few elements attached to that node are selected.Following tables gives the error in the region of maximum stress (at the elementor node where the stress is maximum in the model). The error calculated usingthe two methods is compared with the strain energy error in the model,calculated using ANSYS.

Result Table 1. Element type: PLANE 42No. ofnodes

Nodalsolution:Stress(MPa)

Error(Elementalstressmethod)

Error(Nodalstressmethod)

strainenergyerror

49 93.65 49.78 1.09 14.47683 101.29 41.93 1.19 13.358

101 104.89 32.78 0.01 11.691146 106.95 27.51 0.09 8.88251 108.65 20.80 0.02 8.94491 109.21 15.37 0.00 6.03

1172 109.53 10.08 0.00 3.961342 108.92 7.65 0.00 3.51

Result Table 2. Element type: SOLID 45No. ofnodes

Nodalsolution:Stress(MPa)

Error(Elementalstressmethod)

Error(Nodalstressmethod)

Strainenergyerror

196 107.16 40.48 0.00 10.43210 104.44 38.50 2.33 10.43456 111.28 31.19 1.33 11.58950 112.24 29.05 0.04 7.49

3234 112.88 17.32 0.00 5.164799 113.18 17.25 0.00 4.86

Result Table 3. Element type: SOLID 92No. ofnodes

Nodalsolution:Stress(MPa)

Error(Elementalstressmethod)

Error(Nodalstressmethod)

Strainenergyerror

3198 111.28 42.09 6.48 6.093689 110.84 31.68 4.13 5.094074 111.74 28.82 2.24 2.004151 112.04 18.65 0.97 1.469521 112.34 17.76 0.73 0.77

10527 112.24 17.76 0.43 0.56

From Result table 2, it can be seen that the error is zero calculated by nodalstress method for 196 nodes in the model (column 1). It is because of the stressis maximum at the node belonging one element (not shared by two or moreelements; see observation 2).

Conclusions:1. Error using elemental stress gradient is more accurate method as the elementresults are absolute results (No averaging is done). Also comparing this methodwith nodal stress method this method is error free method and gives fairly gooderror estimation. This method is also suggested in literature (Books) verycommonly.2. Error using nodal stress has some disadvantages and gives zero error whenthe node is not shared by two or more elements, hence care must be takenbefore concluding the error.3. This study does not throw any focus on accepted range of error, but gives fairidea about error estimation. Study can be further extended to get the range ofaccepted error values.4. The methods discussed in the paper gives fair idea about importance of meshconvergence and methods of error estimation. This will help analyst tounderstand and analyze the results of his solution.

REFRENCES:

1. Finite element proceduresKlaus-jurgen BathePrentice-Hall of India printed limited,New Delhi-110001

1997

2. Building better products with finite element analysisVince Adams

3. NAFEMSA finite Element Primer

4. ANSYS 6 Documentation, (Online help).