Reliable FE-Modeling with ANSYS

Transcription

1 Reliable FE-Modeling with ANSYS Thomas Nelson, Erke Wang CADFEM GmbH, Munich, Germany Abstract ANSYS is one of the leading commercial finite element programs in the world and can be applied to a large number of applications in engineering. Finite element solutions are available for several engineering disciplines like statics, dynamics, heat flow, fluid flow, electromagnetics and also coupled field problems. It is well known that a finite element solution is always an approximate solution of the considered problem and one always has to decide whether it is a good or a bad solution. Hence, the question is how a finite element solution can be validated. In this paper we mention several aspects of modelling to obtain a good finite element solution for problems in structural mechanics. Although ANSYS gives quite a lot of warnings to prevent the user from making modelling mistakes, there are still some aspects during the preand postprocessing and also during the solution process we would like to discuss just to point out where user knowledge might still be necessary to obtain good results. In our investigations we use both ANSYS products: ANSYS and ANSYS Workbench. Several ways of modelling a problem of structural mechanics are discussed, taking into account the different available features of ANSYS and ANSYS Workbench. Introduction The finite element method (FEM) is the most popular simulation method to predict the physical behaviour of systems and structures. Since analytical solutions are in general not available for most daily problems in engineering sciences numerical methods have been evolved to find a solution for the governing equations of the individual problem. Although the finite element method was originally developed to find a solution for problems of structural mechanics it can nowadays be applied to a large number of engineering disciplines in which the physical description results in a mathematical formulation with some typical differential equations which can be solved numerically. Much research work has been done in the field of numerical modelling during the last thirty years which enables engineers today to perform simulations close to reality. Nonlinear phenomena in structural mechanics such as nonlinear material behaviour, large deformations or contact problems have become standard modelling tasks. Because of a rapid development in the hardware sector resulting in more and more powerful processors together with decreasing costs of memory it is nowadays possible to perform simulations even for models with millions of degrees of freedom. In a mathematical sense the finite element solution always just gives one an approximate numerical solution of the considered problem. Sometimes it is not always an easy task for an engineer to decide whether the obtained solution is a good or a bad one. If experimental or analytical results are available it is easily possible to verify any finite element result. However, to predict any structural behaviour in a reliable way without experiments every user of a finite element package should have a certain background about the finite element method in general. In addition, he should have fundamental knowledge about the applied software to be able to judge the appropriateness of the chosen elements and algorithms. This paper is intended to show a summary of ANSYS capabilities to obtain results of finite element analyses as accurate as possible. Many features of ANSYS are shown and where it is possible we show what is already implemented in ANSYS Workbench. We will distinguish two different sources of errors within a finite element analysis. On the one hand some mistakes might be introduced in the analysis because the user himself does not know enough about finite elements in general. To minimize these errors we summarize important features of certain element types and element formulations of ANSYS. We also discuss the quality of different element shapes with respect

2 to accuracy and finally provide some information of a correct coupling of different element types. On the other hand errors might occur due to a poor quality of the used finite element program itself. A high-quality program assists the user in reporting reasonable warnings and errors. We will discuss some typical error messages from ANSYS, which allow the user to correct a finite element model immediately. Some reasons possibly leading to poor finite element results are summarized in Tab.1 below to give a coarse overview: Table 1: Possible sources of errors in a finite element analysis Engineer Finite Element Software - wrong element type - bad element formulation Preprocessing - wrong element coupling - bad meshing algorithm - wrong idealization - no warnings and errors - wrong calculation discipline - wrong calculation algorithm Solution - wrong boundary conditions - inaccurate equilibrium iteration - wrong convergence criteria - no warnings and errors - wrong result coordinate system - wrong result averaging Postprocessing - wrong selection of components - wrong displaying of results - wrong interpolation of results - no warnings and errors We should not forget to point out, that ANSYS is a general purpose program, where many numerical modelling techniques are implemented. However, it is sometimes not easy to learn especially for beginners or even for designers, who are usually not finite element experts. With ANSYS Workbench some effort has been done to offer a product, which has implemented by default the best algorithms of ANSYS and is furthermore very easy to use. Hence we will always discuss in this paper as a first step some modelling features of ANSYS and finally point out which of them are already available in ANSYS Workbench. Preprocessing First of all, we will summarize some important aspects every user should be familiar with when doing the preprocessing of a finite element analysis with ANSYS. The following topics will be discussed very briefly: consequences of different element shape functions, important features of different beam and shell elements, results of different element shapes and element formulations and finally correct coupling of different element types. In addition to that, we show what kind of help is available from ANSYS in terms of warnings and errors during the preprocessing to minimize modelling mistakes. User knowledge during preprocessing Choosing an element with linear or a quadratic shape functions To make it as easy as possible we will just look at elements with displacement degrees of freedom no rotational degrees of freedom are present. In ANSYS such elements are called SOLID (acting in 3D), PLANE (acting in D) or LINK (acting in 1D), respectively. Let us consider the one-dimensional case. Within one element the displacements are supposed to vary in a linear or quadratic manner: u( x) a + a x (1) = 0 1 u( x) x = a0 + a1 x + a () Hence, we talk about linear or quadratic elements. As a consequence of that assumption the strain and also the stress distribution is either constant or linear within each element due to:

3 du ε ( x ) = and σ ( x) = E ε( x) (3) dx This may be easily illustrated in the following one-dimensional truss example. The resulting displacement and stress distribution is shown for a two element discretization with linear and quadratic elements: F F 1 u( x) = a0 + a1 x X 1 u( x) = a + a x + a x 0 1 u x Results: u x du σ ( x) = E dx analytic F E M σ ( x) = E du dx σ x σ x Figure 1. One dimensional truss problem: linear and quadratic elements and their consequences With E as Young s Modulus, F as applied force, and A(x) as the cross sectional area the analytic solution is x F 1 u( x) = dx E and A( x 0 ) F σ ( x ) = (4) A( x) coming from a direct integration of the governing differential equation of the above considered problem. It is generally known that better results can be obtained using the same discretization with quadratic or higher order elements compared with the results of linear or lower order elements. You should also keep in mind that the degree of freedom solution always shows a smooth distribution from element to element whereas the distribution of derived quantities (such as strains and stresses) is no longer smooth at the element boundaries. This is a correct result in terms of the finite element method. Just the so-called weak formulation of the concerning differential equation is solved by finite elements and the continuity requirements for the governing variables of the problem are relaxed.

4 Recommendation: ANSYS: ANSYS Workbench: In general, the user should prefer to take a quadratic element if possible. By default, SOLID186/SOLID187 is used as a quadratic 3D Solid-Element Beam elements Bernoulli or Timoshenko Let us now consider a different element type. To model the behaviour of a structure that is thin in two dimensions relative to the third dimension, which is subjected to a bending load, a beam element is good choice. Just the line of gravity of the beam-like-structure has to be discretized. In contrast to classical solid elements beam elements have not only displacement degrees of freedom, but also rotations. A simple supported beam structure under a uniform pressure load distribution is shown in Figure. Y X Figure. Simple supported beam structure under uniform pressure load distribution At this stage it is worth to distinguish two different beam theories: In ANSYS beam elements are available according to the theory of Bernoulli or according to the theory of Timoshenko, respectively. The user should know that shear stresses are not calculated for Bernoulli beam elements but only for Timoshenko beam elements. Bending stresses (linear over the thickness) are available in both beam theories. Since the effect of shear is neglected using Bernoulli beam elements the structure will show a stiffer behaviour as if a Timoshenko beam model was used. This is illustrated in Figure 3. B E R N O U L L I (default for BEAM3 / BEAM4) T I M O S H E N K O (standard for BEAM188 / BEAM189) bending stress σ x bending stress σ x shear stress τ xy shear stress τ xy deformed configuration Y deformed configuration d max_y = X d max_y = Figure 3. Results of a beam structure using Bernoulli beam elements and Timoshenko beam elements It is worth to know about these both theories when using beam elements to choose the correct element type if shear stresses are for example of interest. Considering the deformation it should be mentioned that differences cannot be neglected any more if the ratio of the thickness to the length of the beam - d/l - exceeds the limit of 1/10, i.e. if the beam cannot be considered any more as thin.

5 In [4] the analytical solution for the maximum displacement in y-direction according to Bernoulli is given: 4 5 ql d max_ y = and with q=1, l=1 and EI=17500 d max_ y 384 EI = (5) We observe that the numerical solution with BEAM3/BEAM4 is almost exact. However, since shear stresses are always present in reality, the user has a more accurate result, taking BEAM188/BEAM189. Recommendation: ANSYS: In general, the users should take BEAM188/189 if possible. ANSYS Workbench: By default, BEAM188 is used. Shell elements Kirchhoff-Love or Reissner-Mindlin Shell elements are quite similar to beam elements in the sense that they also have both degrees of freedom - displacements and rotations. Shell elements are typically taken to model a structure subjected to a bending load, that is thin in two dimensions relative to a third. When a thin structure is idealized using shell elements one should know the following: by means of shell elements one actually predicts the structural behaviour of the mid surface of the thin structure. A very easy shell example is shown in Figure 4. Z Y X Figure 4. Shell example: Simple plate subjected to a bending load with a fixed supported edge As for beam elements there also exist two different theories to formulate shell elements. The differences between a typical KIRCHOFF-LOVE shell element like SHELL63 in ANSYS and a typical REISSNER- MINDLIN shell element like SHELL181 is simple: Any shell element based on the KIRCHOFF-LOVE shell theory does not calculate transverse shear stresses. Hence, the resulting deformations may be underestimated, especially in thick shell structures. On the other hand, shell elements based on the REISSNER-MINDLIN theory take into account the shear stress distribution over the thickness. As a consequence of that these elements typically show a softer deformation behaviour due to the presence of shear stresses. As for beam elements the effect of shear deformation can be neglected as long as the shell structure can be considered as slender, i.e. if the ratio of the shell thickness to two typical lengths - d/l 1 and d/l - is less than 1/10. For both shell theories the bending stresses vary linearly with respect to the thickness. The user should know which theory is implemented in which shell element. As an example we analyse the simple shell structure shown in Figure 5. Due to the applied load case, we can check both the correctness of the bending and membrane stiffness. An analytical solution for this problem is derived based on the KIRCHOFF-LOVE theory in [3]. In Figure 6 we compare the analytical result of the radial displacement with the numerical result using SHELL63. We use the default mesh coming from ANSYS Workbench. The numerical solution is really good. Comparing the result of SHELL63 and SHELL181 in Figure 7 it turns out that SHELL181 shows a softer behaviour. This result is the best one, since the effect of shear stresses is also included in the analysis. For the shell problem the following parameters are given: Q=10000, E=10000, µ=0.3, t=15, 0<s<3.61 and θ= From the geometry we can calculate r ϕ and r ϕ0. Note, that r ϕ always depends on the position of s. Note also, that the used finite element mesh is the default mesh the user gets in ANSYS Workbench:

6 s Q r ϕ 0 θ r ϕ r Figure 5. Shell problem: Static system and finite element mesh In [3] we find the analytical solution for the displacements in r-direction depending on the coordinate s: u r () s s λ0 Qsin θ rϕ 0 λ0 rϕ e s 4 0 = cos λ0 with λ0 = 3( 1 µ ) (6) Et r t ϕ 0 r ϕ u r (s=0) = u r (s=0) =.046 Analytical solution Numerical solution Figure 6. Shell problem: Analytical and numerical result of the displacement in r-direction depending on s K I R C H O F F L O V E (standard for SHELL63) R E I S S N E R M I N D L I N (standard for SHELL181) Figure 7. Shell problem: Displacement in r-direction (over-scaled) using SHELL63 and SHELL181

7 Recommendation: ANSYS: ANSYS Workbench: In general, the users should take SHELL181 if possible. By default, SHELL181 is used. Solid elements element technology and element shape Let us consider solid elements again. In this subsection we compare the results of linear and quadratic twodimensional solid elements when modelling a bending dominated problem. We will just focus on the solid elements of the 18x class, i.e. PLANE18/PLANE183. We also have a look at different element shapes. What is mentioned holds also for the three-dimensional case, using SOLID185/SOLID186/SOLID187. For the problem in Figure 8 an analytical solution is available in [4]: Y X B = 1 H = 1 M(F) = 1/6 W = 1/6 σ x = M/W = 1 Figure 8. Simple supported structure subjected to pure bending: geometry model and analytical solution PLANE18 (linear) PLANE18 (linear) PLANE183 (quadratic) σ x σ x σ x Let us first look at the results of quadrilaterals: Using linear elements we can just obtain the correct results using the Enhanced Strain Formulation (instead of the Full Integration Technology). This method prevents the elements from shear locking. This formulation should be activated by the user if it is known from the beginning of the analysis that the problem will be bending dominated. Using quadratic elements results in a correct bending stress distribution over the thickness. Enhanced Strain Formulation Figure 9. Results of bending stresses for linear and quadratic elements using a mesh of quadrilaterals PLANE18 (linear) PLANE183 (quadratic) σ x σ x We compare the results of quadrilaterals from Fig. 9 with the results of triangles in Fig. 10: It is important to realize that linear elements are not able to predict the correct results well when they are used as triangles even if the Enhanced Strain Formulation is used. Using quadratic elements the results are correct even when triangles are used. Figure 10. Results of bending stresses for linear and quadratic elements using a mesh of triangles

8 Linear triangles never should be used whereas linear quadrilaterals can be used without any problems in a structural analysis. If the problem is bending dominant the Enhanced Strain Formulation should be activated. Quadratic triangles and quadrilaterals are always a good choice. The same thing is also valid for the three-dimensional case. Recommendation: ANSYS: In general, the users should take PLANE183, SOLID186/SOLID187 if possible. ANSYS Workbench: By default, SOLID186/187 is used. Coupling of different element types In this subsection we focus on coupling of different element types, especially those who have not the same degrees of freedom. As an example we consider the following little example in Figure 11, where a beam or shell element (displacements and rotations) is coupled with solid elements (displacements): Solids Beam Shell = joint Incorrect coupling of elements with different degrees of freedom: Unintentionally a joint is created at the couple point. A rigid body motion is possible. The resulting stiffness matrix will become singular and the problem cannot be solved like this. Figure 11. Incorrect coupling of elements with different degrees of freedom However, there are a few possibilities to correctly transmit the beam s or shell s rotation into the solid part of the structure. Two finite element models are shown below in Figure 1, where additional beam or shell elements are used to perform the coupling reasonably: Beam Shell Beam Shell Solids Solids Figure 1. Correct element coupling using additional finite elements It is obviously that in both finite element models from Figure 1 the originally modelled joint from Figure 11 no longer exists and that the rotations are transmitted correctly by some additional elements. A quite new technique to solve the above problem is given via the MPC (Mulit Point Constraint) method:

9 Beam Shell MPC Solids Figure 13. Correct element coupling using the MPC method Using the MPC technique, ANSYS generates internally some coupling equation s to establish the correct kinematics at the coupling point. In fact, the MPC technique is valid if a small and also a large deformation analysis is performed. MPC is modelled via bonded contact together with a special contact solution algorithm. The MPC technique is also already available in ANSYS Workbench. Help from ANSYS during preprocessing Element shape function and element shape It was outlined above that linear elements should not be used as triangles or tetrahedra. However, if such elements are still generated, ANSYS will give the following warnings from Figure 14. The user has immediately the chance to correct the model to obtain better results. Since in ANSYS Workbench only quadratic solid elements are available those warnings are not necessary in ANSYS Workbench: Figure 14. ANSYS warnings to avoid linear elements as triangles or tetrahedra Identification of rigid body motions in a structure We have discussed above that a rigid body motion may be modelled in a structure due to an incorrect coupling of different element types. Since the resulting stiffness matrix is singular ANSYS will not be able to solve this problem. The error messages from Figure 15 appears in ANSYS and ANSYS Workbench:

10 ANSYS ANSYS Workbench Figure 15. ANSYS and ANSYS Workbench messages due to incorrect element coupling (rigid body motion) However, it is also possible to introduce a rigid body motion into the system, if the structure is not supported in a statically determined way. Such a situation is modelled in Figure 16 and it is shown how ANSYS will identify such a modelling mistake, which has then to be corrected by the user: Figure 16. ANSYS error messages to identify rigid body motions due to missing supports Note, that ANSYS Workbench generates weak springs, if the program recognizes that a system is not supported in a statically determined manner. Adding weak springs prevents the system from a rigid body motion. If such a situation occurs, ANSYS Workbench gives the following information on screen: Figure 17. ANSYS Workbench information to identify rigid body motions due to missing supports

11 Automatic shape checking control A topic we did not discuss so far in this paper is the resulting mesh quality after discretizing the structure with finite elements. It is well known that the accuracy of results within one element also depends on its shape. Fortunately, there are a few algorithms implemented in ANSYS to check the quality of the resulting element shapes to minimize the errors due to bad element shapes. For details we refer to [1]. If one of the shape checking criteria is not satisfied the user will get a warning in ANSYS like shown in Figure 19. Furthermore, it is possible to indicate the bad elements in the mesh. So it is easy to identify regions where remeshing is required. To illustrate this we recall again the simple bending problem to compare the numerical results with the analytical ones. In Figure 18 it is indicated that we expect a maximum bending stress of σ x = 1: Y X B = 1 H = 1 M(F) = 1/6 W = 1/6 σ x = M/W = 1 Figure 18. Simple supported structure subjected to pure bending: geometry model and analytical solution In the following Figure 19 we show the results of a bad discretization using linear elements. The correct results have already been shown in Figure 9 and Figure 10. We recognize, that especially within the indicated bad elements the numerical results are poor. The shape warnings coming from ANSYS are also shown for this case and the user has immediately the chance to correct the mesh: PLANE18 (linear) σ x Figure 19. ANSYS warnings to identify bad shaped elements and numerical results of bad elements At this stage we should note, that shape warnings will not be shown in ANSYS Workbench. However, since ANSYS Workbench just uses quadratic solid elements this is not really a disadvantage, since those elements are not as sensitive with respect to the resulting element shape. The above warnings would not have been appeared if PLANE183 (quadratic) would have been used instead of PLANE18 (linear).

12 Solution Solving a linear problem of structural mechanics meanwhile has become a standard task and can be performed without any difficulties. The accuracy of the results just depends on the quality of the resulting finite element model itself. By contrast a nonlinear problem has to be solved iteratively and a solution is only obtained if convergence is achieved. At this stage we would like to give a certain background about the solution of nonlinear problems. First of all, we summarize briefly the basics of the iterative solution method which is implemented in ANSYS to solve nonlinear problems. With this knowledge it should be possible to understand how certain user settings on convergence criteria might influence the accuracy of a nonlinear solution. In the last section we talk about the necessity of a geometric nonlinear calculation. We show what kind of error is introduced in the analysis if the effect of large deflections is ignored numerically. We try to show the limit of a geometric linear calculation considering the effect of large deflections. General remarks about the solution of nonlinear problems Every linear problem in structural mechanics results in the solution of the matrix equation K u = F, where K is the stiffness matrix, u denotes the vector of displacements / rotations and F is the vector of forces / moments. On the other hand in a nonlinear problem of structural mechanics the resulting matrix equation can be formulated as K(u) u = F and it should be realized that the stiffness matrix depends on the displacements / rotations. Due to this fact an iterative solution scheme is required. In ANSYS the Newton-Raphson method is implemented to solve a nonlinear problem iteratively. The iterative solution process can be described by the following equations together with Figure 0. K u F F T NR = with F F NR = R (7) In every Newton-Raphson iteration the changing stiffness can be identified as the slope of the force deflection curve from Figure 0. Hence we speak about a tangential stiffness K T. In every iteration a displacement / rotation increment u is calculated until the imbalance forces which are also called the residual forces / moments R become acceptable small. Strictly speaking, a structure is only in equilibrium if the residual forces /moments totally vanish. In the current version of ANSYS (8.0) the user has the chance to postprocess the residual forces / moments to check the accuracy of the simulation. F Detailed consideration of the iteration: F R criterion (small number) u u u Equilibrium is obtained if: R < criterion Figure 0. The Newton-Raphson iteration scheme

13 To be able to compare the vector of residual imbalance forces or moments with a single scalar value a norm of the vector is needed. Different vector norms are available in ANSYS and we refer to []. Help from ANSYS during the solution of nonlinear problems ANSYS shows during a nonlinear solution a convergence monitor where the numerical behaviour during the Newton-Raphson iteration can be studied. Hence, the user can observe whether the solution process shows either a converging trend or an unstable behaviour. The iterative process which is graphically displayed in the convergence monitor is also documented in the Output Window. If the iterative process does not show a converging trend the user can stop the nonlinear iteration, think of certain techniques to overcome the convergence difficulties and start the analysis again. ANSYS Workbench has not yet implemented the convergence monitor. However, the Output Window of ANSYS is available where the user is able to study the iterative solution procedure as well. Iteration control of the residual forces and moments By default for elements with displacement degrees of freedom the imbalance residual forces are iterated below a certain criterion. However, if an element type supports rotational degrees of freedom, too, the imbalance residual moments are also corrected in every iteration until they fall under a certain limit. Figure 1 shows a typical convergence diagram where the iteration process can be studied: Convergence is obtained, if the norm of the residual force and moment vector (denoted by F L / M L) falls below the criterion (called F CRIT / M CRIT). The considered structure is then in equilibrium!!! The value for the criterion is calculated by ANSYS but can be overwritten by the user if desired. Figure 1. The Newton-Raphson iteration of the imbalance residual forces and moments Iteration control of the incremental displacements and rotations It should be noted, that it is also possible to observe the evaluation of the incremental displacements and rotations from iteration to iteration. Taking again a norm of the corresponding vectors allows the user to stop with the iteration process if this norm is acceptable small. Hence, it is possible not to measure the norm of the imbalance forces and moments but to stop with the iteration process as soon as the norm of the incremental displacements and rotation falls under a certain limit. Such an iteration is show in Figure.

14 The iteration stops, if the norm of the incremental displacements (U L) and rotations (ROT L) falls below the given criterion (U CRIT / ROT CRIT). The considered structure is then not necessarily in equilibrium!!! The value for the criterion is calculated by ANSYS but can be overwritten by the user if desired. Figure. The Newton-Raphson iteration of the incremental displacements and rotations Note carefully, that in case of activating only the incremental displacement and rotation control, you should always check the results for equilibrium. In Figure 3 it is illustrated that especially for structures with a stiffening behaviour a small increment in the displacements does not necessarily mean that also the imbalance residual forces are already acceptable small. If there are still certain residual forces in the system (imbalance forces) the structure is obviously not yet in equilibrium. It can easily be observed that the incremental displacement u might already be acceptable small but the imbalance residual force R might not. F R u u Figure 3. Situation where just a displacement controlled iteration might lead to erroneous results The user should know that when activating the incremental displacement control an additional equilibrium check should be performed after convergence has been achieved. It is also possible to specify individually for which quantity the evaluation of the increments during the Newton-Raphson iteration should be measured.

15 About the necessity of a geometric nonlinear calculation In this section we discuss the limit of a geometric linear calculation. We will show that with the classical definition of strain like it is used within a geometric linear calculation it is not possible to calculate a rigid body rotation in a correct manner. However, by introducing a new nonlinear strain definition we are able to calculate correct results. It will be shown that for large rigid body rotations the geometric linear theory looses its validity. Let us focus on the following simple rod problem in Figure 4. The rod rotates as a rigid body with rotation angle ϕ. Point A will move to point A. The components of its movement are also calculated in Figure 3 in an analytical manner: y, v A u ( x) = x(cosϕ 1) v ( x) = x sinϕ ϕ A x, u Figure 4. Single rod with rigid body rotation: system sketch and analytical results of rigid body kinematics In Eq. (3) the definition of engineering strain in x-direction was given for a geometric linear calculation. Substituting the results from kinematics from Figure 4 we obtain the strain depending on ϕ: u ε( x ) = = cosϕ 1 (8) x Actually we expect a zero strain component in x-direction for the above problem. In the following Tab. we show that this holds only approximately for very small angles ϕ. We think that the geometric linear theory looses its validity if rotation angles are bigger the 10 : Table : Resulting strain of a rigid body rotation using a geometric linear calculation ϕ ε -0,000-0,0038-0,015-0,99 One idea to solve the problem from above is to define the strain in a different way, i.e. using a different strain measurement, like it is given for example in Eq. (9) taken from [5]. This definition of strain is called Green-Lagrange strain. It is a nonlinear strain measurement with respect to the displacements: u 1 u 1 v 1 ( ) 1 ε ( x ) = + + = cosϕ 1+ cosϕ 1 + ( sinϕ ) (9) x x x

16 Using this strain measurement results in zero strain for all angles ϕ so a reasonable correct result is calculated. To use this definition of strain a geometric nonlinear analysis is required. Table 3: Resulting strain of a rigid body rotation using a geometric nonlinear calculation ϕ ε Recommendation: ANSYS: If the user is not sure to include or exclude the geometric nonlinear effects in the simulation, a geometric nonlinear calculation is for sure always the better choice. However, if a linear calculation has already been performed and the resulting strains are small for example below % the error in the analysis will be small and acceptable. If more than 5% strain is indicated a geometric nonlinear analysis should be performed to obtain better results. ANSYS Workbench: By default, ANSYS Workbench gives the following information if the effect of geometric nonlinearities should be included in the analysis: Figure 5. ANSYS Workbench information to activate geometric nonlinearities Postprocessing In this chapter we first summarize basic knowledge the user should have considering postprocessing especially in ANSYS. Most features are not yet available for ANSYS Workbench. User knowledge during postprocessing Nodal and element solution It follows directly from the theory of finite elements that discrete values of the degrees of freedom are calculated and available only at the nodes of the model. Hence, they can only be displayed with the nodal solution option of ANSYS. Derived quantities - such as element strains and stresses - are calculated at the integration points which are located somewhere inside each element and values are extrapolated to the nodes. Strains and stresses should first be displayed with the element solution option of ANSYS. The nonsmoothness of the strain and stress field can only be recognized if strains and stresses are shown with the element solution option. However, it is possible to postprocess strains and stresses also with the nodal solution option which simply means that the extrapolated element values of the integration points are averaged at the nodes. This is illustrated in Figure 6.

17 Node Integration points Calculation of the values of derived quantities such as strains and stresses at the nodes: Value Node = Elem. Value Element Elem. Figure 6. How element quantities like strains and stresses are calculated at the nodes What we have outlined so far should be illustrated by a little example shown in Figure 7. A quarter of a plate with a hole is modelled by linear elements and subjected to a traction force. Only one material is used. The contour plots show the stresses in x-direction, displayed with the element and nodal solution option: Y X System Element solution: σ x (not averaged) Nodal solution: σ x (averaged) Figure 7. Results from the element and nodal solution in ANSYS It can be recognized that the stress field is not smooth in the element solution as it is characteristic for finite element solutions. Displaying the same quantity with nodal solution results in a smooth contour distribution due to the averaging process. The element solution is useful to identify high result gradients within single elements. In those areas a finer mesh may be required. A second very similar example is shown in Figure 8. This time the structure is made up of two different materials a soft material is combined with a stiff material. Clearly, it is not allowed to average the element stresses at those nodes where elements with different materials touch each other. In ANSYS this is recognized automatically if the postprocessing is done using the powergraphic option. If the full graphic option is used all element values at the nodes are averaged even at the material boundary which can be misleading. The powergraphic option is activated by default in ANSYS.

18 soft stiff Y X System Element solution: σ x (not averaged) Nodal solution: σ x (averaged) Nodal solution: σ x (averaged) FULL GRAPHIC POWERGRAPHIC Figure 8. Results from element and nodal solution using full graphic and powergraphic in ANSYS Help from ANSYS during postprocessing Estimating the discretization error: The energy error approach One possibility to estimate the discretization error in ANSYS is using the implemented energy error approach. The basic ideas of this concept are taken from [6]. At this stage we will just very briefly describe how the total energy error of a selected domain of the structure is calculated: At every node n of an element i the following quantity can be computed: σ = σ σ with i n a n i n a σ n averaged stress at the node (10) i σ n extrapolated stress to the node This quantity is taken to calculate the total energy error of the structure according to: e E = 100 with U total elastic strain energy in the selected domain (11) U + e e total energy error in the selected domain Elem 1 e = i= 1 V i T -1 i { σ } D { σ } dv In other words the total energy error weights the different σ n i using an energy formulation to obtain a global measurement of the discretization error for the selected domain. It is calculated in percent. e n n

19 Estimating the discretization error: Comparing the extrapolated and averaged stresses Recall equation (10): σ n i = σ n a σ n i. An easy way to estimate the discretization error is to take the maximum value of the computed nodal quantity σ n i in each element and define it as the characteristic value for each element i. This local measurement of the discretization error is called SDSG: SDSG = Max { σ n i } (1) Example of measuring the discretization error The two outlined possibilities to measure the discretization error will be demonstrated in a little example. The system with boundary conditions and the analytical solution is shown in Figure 9. Y X A R =1 F=0 σ L = σ R =0 A L =10 Figure 9. System with analytical solution to be compared with different finite element discretizations In Figure 30 the resulting stress distribution in x-direction is shown together with the local element quantity SDSD and the global error energy to measure the discretization error: σ x σ x SDSG SDSG E = % E = % Figure 30. Results of measuring the discretization error locally and globally in ANSYS Classic

20 Clearly, the right discretization in Figure 30 is better than the left one. However, only postprocessing SGSG gives one an idea in which regions the mesh should be refined. The total error energy just indicates how good or bad a discretization is in general. Unfortunately, both methods to measure the discretization error just work for linear static analyses. Adaptive methods to minimize the discretization error With an adaptive method you are able to minimize the error in a finite element analysis step by step running a few analyses one behind the other. The following adaptive methods are implemented in ANSYS: p-method (ANSYS): The idea of the p-method is to use higher order shape functions within the elements. However, the discretization of the problem will be unchanged, i.e. for all analyses the same mesh is used. You may for example start with an analysis using linear elements, followed by one with quadratic elements, cubic elements and so on. Up to eight-order-polynomials may be used as shape functions to formulate finite elements. h-method (ANSYS and ANSYS Workbench): The idea of the h-method is to refine the mesh automatically in regions where certain results are of interest. The order of the elements will remain unchanged, just an adaptive discretization is performed. In Figure 31 we show the results of using the p-method and the h-method, respectively. The result quantity, which is of interest, may be a characteristic stress. It can be recognized, that both methods iteratively improve the accuracy of the obtained results: p-method: 5 σ 4 h-method: Order of the shape function σ Loop-1 Loop-3 Loop-1 Loop- Loop-3 Loop-4 Mesh refinement loop Figure 31. Adaptive methods in ANSYS and ANSYS Workbench

21 Evaluating the residual forces / moments in a nonlinear analysis Since for most nonlinear problems analytical solution are not available it is difficult to validate a nonlinear numerical result coming from the finite element method. But what the user always can do is to check the quality of the state of equilibrium in the system. Remember the third chapter of this paper: Here we noted that during the nonlinear solution process the imbalance residual forces / moments are iterated to become acceptable small. Numerically they will never be perfectly zero. So a reasonable question would be how big the residual forces are in the end of a converged nonlinear solution. Clearly, the residual forces / moments should be closed to zero. Within ANSYS in its actual version (8.0) the user has indeed the chance to postprocess the residual forces / moments. An example is shown in the following Figure 3. Figure 3. Residual forces in x-direction of a converged nonlinear solution step in ANSYS What we see is a contour plot of the residual forces in x-direction of the last converged solution step of a nonlinear analysis. It can be recognized that very locally there is a certain amount of disequilibrium in the structure. However, if the user wants to improve the accuracy of the nonlinear result it is possible to iterate the residual forces / moments to a smaller value. Note, that the residual forces / moments can be displayed for every equilibrium iteration within the Newton-Raphson iteration. Hence, is possible even to judge about the state of equilibrium or disequilibrium in a solution step which has not yet converged. Conclusion In this paper we tried to discuss several modelling aspects to keep in mind when doing a finite element analysis with ANSYS and ANSYS Workbench to improve the accuracy of the results. For the three typical steps within a finite element analysis the preprocessing, the solution and the postprocessing we provided some basic finite element knowledge the user should have to be able to build up a reasonable finite element model for the considered problem. Considering the preprocessing we discussed the finite element results of different element types, element shapes and element orders. If available, numerical solutions have been compared with analytical results. We also described briefly the correct element coupling of different element types. In addition to that, we showed intelligent warning messages of ANSYS and ANSYS Workbench to minimize modelling mistakes. Especially the solution of nonlinear problems requires some knowledge about the iterative solution scheme of ANSYS and ANSYS Workbench. In this paper we discussed briefly the background of the Newton- Raphson-Method. The limit between a geometrical linear and nonlinear calculation has also been shown. Finally we mentioned some aspects of the powerful postprocessing of ANSYS and ANSYS Workbench. Several features are available to check every finite element result for its numerical correctness like error estimators or using adaptive finite element methods.

22 References [1] ANSYS Elements Reference, Release 6.1, Swanson Analysis Systems, Inc., 001 [] ANSYS Theory Reference, Release 6.1, Swanson Analysis Systems, Inc., 001 [3] Pflüger, A., Elementare Schalenstatik, 5. Auflage, Springer-Verlag, 1981 [4] Schneider, K.-J., Bautabellen für Ingenieure, 11. Auflage, Werner-Verlag, 1994 [5] Wriggers, P., Nichtlineare Finite-Element-Methoden, Springer-Verlag, 000 [6] Zienkiewicz, O.C., Zhu, J.Z., A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis, International Journal for Numerical Methods in Engineering, Vol. 4, pp , 1987