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1 Computers and Structures 89 (2011) Contents lists available at ScienceDirect Computers and Structures journal omepage: Improved stresses for te 4-node tetraedral element Daniel Jose Payen, Klaus-Jürgen Bate Department of Mecanical Engineering, Massacusetts Institute of Tecnology, Cambridge, M 02139, US article info abstract rticle istory: Received 12 December 2010 ccepted 14 February 2011 vailable online 31 Marc 2011 Keywords: Stress improvements Stress recovery es Nodal point forces Finite elements 4-Node tetraedral element Te objective in tis paper is to present te metod for te calculation of improved stresses publised by Payen and Bate in [1] for te 4-node tree-dimensional tetraedral element. Tis element is widely used in engineering practice to obtain, in general, only guiding results in te analysis of solids because te element is known to be poor in stress predictions. We sow in tis paper te potential of tis novel approac to significantly enance te stress predictions wit te 4-node tetraedral element at a relatively low computational cost. Ó 2011 Elsevier Ltd. ll rigts reserved. 1. Introduction Most engineering problems in solids and structures are treedimensional in nature. Since te geometry and oter data of te problem are ten usually complex, te structure is best analysed using finite element metods. Te crucial step in any finite element analysis is to coose an appropriate matematical model for te pysical structure (or more generally te pysical penomenon), since a finite element solution solves only tis model, see Ref. [2]. For example, if te structure is tin in one direction and long in te oter two directions a sell matematical model is appropriate, and te problem is solved efficiently using te MITC sell elements, see Refs. [3 6]. However, if te lengt scales of te structure are similar in all directions, and te loading is general, ten tere is no option oter tan to solve te problem using an assemblage of discrete tree-dimensional solid elements, see Refs. [2,7]. Te simplest tree-dimensional solid element available to te finite element analyst is te 4-node constant strain tetraedral element. Tis element is used abundantly in practice because te analyst is able to mes almost any volume regardless of complexity, te element is robust in contact analysis, te element matrices are inexpensive to calculate, and te resulting global stiffness matrix as a relatively small bandwidt. In a typical approac, te analyst would use a mes of 4-node tetraedral elements, in a first analysis, to identify te locations of ig stress concentrations, and ten based upon tese results, te analyst would refine te mes Corresponding autor. address: kjb@mit.edu (K.J. Bate). or, if possible, convert te mes to 10-node or 11-node tetraedral elements in te localised regions of concern, see Ref. [7]. Tis is necessary, simply because te stresses predicted using te 4-node tetraedral element are known to be poor, and te lack of accuracy can be seen using stress band plots of unsmooted stresses, see Refs. [2,8]. Our objective in tis paper is to apply te metod publised by Payen and Bate in [1] to te 4-node tree-dimensional tetraedral element, and sow tat by using a simple algoritm, we are able to enance te stresses in localised regions of concern, witout aving to refine te mes or re-analyse te model. Wile we focus in tis paper on linear static analysis and smoot stress conditions, te results are fundamental and migt be used also in dynamic analysis and nonlinear solutions [2,9,10]. Te teory used for te metod as been publised in detail in Ref. [1], and ence we sall only summarise te fundamental equations and teir properties in tis paper. Te stress prediction is based on te fact tat te element nodal point forces are of iger quality tan te directly-calculated finite element stresses [2,7]. Hence we use two principle of virtual work statements involving tese nodal forces, as summarised in Section 2, to calculate te improved finite element stresses. Indeed, te special properties of te element nodal point forces ave been known for many years; owever, relatively little attention as been given to teir use to improve te finite element stress predictions, see Refs. [1,11,12] and te references terein. In Ref. [1] we mention wy our approac is more general and powerful tan tose previously considered. Wen performing a properly formulated finite element solution, two important facts old, namely, (1) at eac node, te sum of te /$ - see front matter Ó 2011 Elsevier Ltd. ll rigts reserved. doi: /j.compstruc

2 1266 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) element nodal point forces balances te externally applied nodal point loads, and (2) eac element is in force and moment equilibrium under te action of its own nodal point forces, irrespective of te coarseness of te mes, see Refs. [2,7]. For tis reason, it seems somewat natural to use tese forces to calculate improved stress predictions but te details of establising a general and effective algoritm are far from apparent. We named te procedure given in Ref. [1] te nodal point force based stress calculation metod or te NPF-based metod giving es, for sort. In Ref. [1] we sowed tat te NPF-based metod can be used effectively to significantly improve te accuracy of te finite element stress predictions obtained using te 3- and 4-node displacement-based elements in two-dimensional analyses. It is reasonable to expect similar improvements for te 4-node tree-dimensional tetraedral element, and our objective erein is to present a detailed procedure towards tat aim. We solve te same set of problems considered in Ref. [1], but of course tis time in tree-dimensional settings. s expected, we see a significant improvement in te accuracy of te stress predictions for all problems considered. Tese results are of particular interest, since reliable improvements in stresses for te 4-node tetraedral element, using incompatible modes or enanced strains, are difficult to reac in general analyses [13 15]. Fig. 1. Te stress calculation domain for te 4-node tetraedral element; element m would be te central element or a periperal element. Fig. 2. Five test problems for te 4-node tetraedral tree-dimensional element (E = 72E9, m = 0.0, p = 100, F = 6,000, t = tickness): (a) te beam in pure bending problem, (b) te finite plate wit a central ole under tensile loading problem, (c) te square cantilevered plate under sear loading problem, (d) te curved structure in pure bending problem, and (e) te tool jig problem.

3 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) Fundamental equations of te metod detailed review of te general and well-known principles used in te nodal point force based stress calculation metod is given in Ref. [1], and ence we sall only summarise ere te fundamental equations used. In Section 3 we ten focus on te specific details to apply tese principles and teir properties in order to establis improved stress predictions for te 4-node tree-dimensional tetraedral element. We assume linear analysis conditions and use te notation of Ref. [2]. s is standard, we establis te stiffness matrix in te usual manner, solve for te element nodal point displacements U, and te directly-calculated finite element stresses are ten given by s ðmþ ¼ C ðmþ e ðmþ ¼ C ðmþ B ðmþ U ð1þ were C ðmþ, e ðmþ, and B ðmþ are te stress strain matrix, te finite element strain vector, and te strain displacement matrix of element m, respectively. Te element nodal point forces F ðmþ corresponding to te directly-calculated finite element stresses s ðmþ are defined as Z n o F ðmþ ¼ s ðmþ dv ð2þ V ðmþ B ðmþt were V (m) is te volume of element m. If we assume tat tere exists and we can calculate improved finite element stresses s ðmþ from tese element nodal point forces, we obtain two fundamental equations involving tese unknown stresses (and ence te unknown coefficients used to express s ðmþ ). Te first fundamental equation states tat for any virtual displacement field contained in te element interpolation functions, te virtual work by te element boundary tractions is equal to te virtual work by te element nodal point forces (adjusted for body force effects), and ence we call tis equation te principle of virtual work in te form of boundary tractions Z S ðmþ f H ðmþt Z s ðmþ n ðmþ ds ¼ F ðmþ H ðmþt f B dv V ðmþ were H ðmþ, S ðmþ f are te displacement interpolation matrix and te total external surface area of element m, respectively, and n ðmþ is te unit normal to te element boundary. In te absence of body forces f B, Eq. (3) reduces to Z ds ¼ F ðmþ S ðmþ f H ðmþt s ðmþ n ðmþ Te second fundamental equation states tat for any virtual displacement field contained in te element interpolation functions, te element internal virtual work is equal to te virtual work of ð3þ ð4þ Regular mes -2 (-102%) 71 (-29%) Irregular mes 14 (-86%) 79 (-21%) Fig. 3. Longitudinal stress results for te beam in pure bending problem. Te solution error is given in te parenteses.

4 1268 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) te element nodal point forces, and ence we call tis equation te principle of virtual work in te form of internal stresses Z V ðmþ B ðmþt s ðmþ dv ¼ F ðmþ were, of course, te element nodal point forces F ðmþ always correspond to te directly-calculated finite element stresses s ðmþ, see Eq. (2). We sould note tat in Eqs. (3) and (5), te s ðmþ are assumed stress fields over te element m and ence we do not only use unknown tractions between finite elements. Te NPF-based metod uses, as its ingredients, tese two fundamental virtual work statements Eqs. (3) and (5) to obtain finite element stresses tat we can expect to be more accurate tan tose given by Eq. (1). We expect tat, in general, more accurate stresses are predicted because, firstly, te metod allows us to assume a ricer functional space for te stresses tan tat implicitly assumed in establising te stiffness matrix, and, secondly, te nodal point forces are used wic always satisfy te above-mentioned important equilibrium requirements, irrespective of te coarseness of te mes. However, we do not ave a proof tat te stresses will always be improved at a particular location of te model. ð5þ 3. Improving te stresses of te 4-node tetraedral element In order to establis improved stress predictions for a general finite element m, te calculation algoritm employs four basic steps: 1. Solve, in te usual manner, for te element nodal point displacements U, and te element nodal point forces F ðmþ, in accordance wit Eq. (2). 2. ssume appropriate functions for s ðmþ across a predetermined patc of elements; we call tis patc of elements te stress calculation domain. 3. Use te two principle of virtual work statements Eqs. (3) and (5) to solve for te unknown stress coefficients in s ðmþ. 4. Finally, to establis te improved stresses for an individual element m, te stress coefficients corresponding to all possible element combinations to obtain stress calculation domains tat contain element m are calculated using te above steps, and te results are averaged for element m. Of course, it is important to select appropriate functions for te stress fields in s ðmþ, since we aim to ave a sufficiently ric Mes (-33%) 374 (-15%) 81 (-55%) 111 (-38%) Mes (-28%) 402 (-9%) 75 (-58%) 124 (-30%) 362 (-18%) 431 (-2%) 110 (-38%) 184 (+4%) Fig. 4. von Mises stress results for te finite plate wit a central ole problem. Te solution error is given in te parenteses.

5 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) assumed stress space for te stress calculation domain. Clearly, te dimension of te assumed stress space must depend on te number of elements used witin te stress calculation domain. Tat is, for a given dimension of assumed stress space, we must ave tat te domain contains a sufficient number of elements, suc tat te problem solution for te unknown stress coefficients is well-posed for all possible domain geometries tat migt be used. In te specific case of te 4-node tetraedral tree-dimensional element, we assume te stresses to be linearly interpolated and continuous across te entire stress calculation domain, s ðmþ ij ¼ a ij 1 þ aij 2 x þ aij 3 y þ aij 4z for m ¼ 1; 2; 3; 4; 5 ð6þ were te (i, j) refer to te coordinate directions, and te a ij k are te twenty-four unknown stress coefficients to be found. s an aside, we note tat for te 3-node constant strain triangle considered in Ref. [1] we instead assumed bilinear interpolations across its stress calculation domain. Wit te assumption in Eq. (6), eac stress calculation domain for te 4-node tetraedral element sall contain at least five elements, tis way we ensure a well-posed problem for te solution of te coefficients. ltoug any five adjacent elements could be used, we define a stress calculation domain in a quite natural manner as te unique combination corresponding to a central element surrounded by four periperal elements, were eac periperal element sares a face wit te central element, as sown in Fig. 1. Tis stress calculation domain allows us also to maximise te accuracy of te stress prediction, since te averaging in step 4 is used, see above and te furter comments below. In general, te algoritm solves for te unknown stress coefficients in s ðmþ by imposing Eq. (3) to all possible closed contour boundaries contained witin te stress calculation domain, and in addition Eq. (5) to te complete domain. However, in tis case, we ave assumed te stresses to be linearly interpolated, and ence we need to only apply Eq. (3) in order to solve for te stress coefficients. Te reason is tat in te absence of body forces, Eq. (5) is not independent of Eq. (3), see Ref. [1]. Furtermore, we assume inter-element stress continuity, and ence Eq. (3) can be imposed to every possible closed contour boundary by simply imposing te equation to te five tetraedral element boundaries. Mes 1 Mes 2 Fig. 5. In-plane sear stress results for te square cantilevered plate problem across section.

6 1270 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) In tis way, we generate sixty equations, of wic, for te configuration considered in Fig. 1, only tirty-tree are linearly independent. Since tere are twenty-four unknown stress coefficients, te system of equations is over-determined, and so, in general, a solution wic exactly satisfies Eq. (3) does not exist. Hence we use te least squares metod to solve for te unknown stress coefficients, wit te consequence tat te element nodal point forces calculated from te NPF-stresses (see Eq. (4)) will only satisfy te individual element and nodal equilibrium properties mentioned earlier, in a least squares sense. Finally, to obtain te improved stresses for eac tetraedral element m, we average te stress coefficients corresponding to te possible stress calculation domains tat contain element m. Of course, for te cosen geometry tere can be no more tan five domains tat contain element m, tat is, respectively, one and four domains for te element taking te position of te central element and te periperal elements. In te exceptional case tat no domain, as described above, exists wic contains element m (e.g. in a corner of a mesed geometry), we simply construct te stress calculation domain using four elements tat are properly connected to element m, and no averaging is applied. Since we assume te stresses to be linearly interpolated, te numerical effort involved in improving te stress predictions for eac tetraedral element is given by te effort required to solve for twenty-four unknown stress coefficients at most five times (tat is, we must calculate te stress coefficients corresponding to every possible domain wic contains element m). Tis computational effort is relatively small, but, also, an important feature of te algoritm is tat tere is no need to apply tese stress calculations to all elements in te assemblage, instead only to tose elements were improved stresses sould be calculated. Indeed, in practice, te finite element analyst is not always able to perform due to stringent constraints on time and computational resources a detailed mes refinement stress convergence study, especially for complex problems tat are expensive to solve. Instead, in many cases, te analyst will solve te problem only once, using te finest mes possible tat for te available computational resources still results in a reasonable solution time. Given tis solution and te above rater simple algoritm, it is ten possible to enance te stress prediction wit relatively little computational effort in only te specific areas of concern. In addition to enancing te stress prediction, te results obtained wit te algoritm give, of course, also insigt into te accu- Mes 1 Mes 2 Fig. 6. Longitudinal stress results for te curved structure problem across section.

7 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) racy of te es. Namely, if te two stress values are far apart, in important areas of te model, te mes used is too coarse for te es to be sufficiently accurate. We recognise tat we ave not matematically proven stability of te algoritm for all possible geometries of te stress calculation domains. Terefore, it is possible, tat for certain meses wit grossly distorted elements te algoritm establises ill-conditioned matrices in wic case te solution would ave to be abandoned for tat particular domain (were te elements are too distorted). However, we ave tested te procedure in a large number of domains containing igly distorted elements and ave not encountered tis difficulty. Hence our experience is tat as long as te mes is reasonable (wic is anyways required for te original displacement solution) te algoritm seems to be quite robust and stable. Te effectiveness of te algoritm for te 4-node tetraedral element is illustrated using te same five test problems as considered in Ref. [1]: a beam in pure bending, a finite plate wit a central ole under tensile loading, a square cantilevered plate under sear loading, a curved structure in pure bending, and a tool jig problem (like considered in Ref. [7]). We define tese test problems in Fig. 2, and sow te results (rounded to full digits) in Figs. 3 8 respectively, were te refers to te stresses calculated using te proposed nodal point force based stress calculation metod. Considering tese results, te values given in te band plots are un-averaged, wile te given numerical stress values are te averaged nodal point values wit te solution error sown in parenteses. Tis error is measured wit respect to te solution (called exact in figures) obtained using a very fine mes of 27- node exaedral elements. Note tat a given numerical stress value may be outside te scale of te band plot because we selected te scale to reasonably indicate te stress variation over te complete domain. s expected, we see a significant improvement in te accuracy of te predicted stresses for all problems solved. However, te improvement in stresses is somewat less tan wat we ave seen for te 3-node constant strain triangle in Ref. [1], wic is partly due to te fact tat, for te tree-dimensional analyses, we are using linear, and not bilinear, stress interpolations, see Eq. (6). Mes (-75%) 188 (-69%) 387 (-31%) 351 (-42%) Mes (-61%) 283 (-54%) 453 (-20%) 475 (-22%) 331 (-41%) 401 (-34%) 579 (+3%) 595 (-2%) Fig. 7. von Mises stress results for te tool jig problem. Te solution error is given in te parenteses.

8 1272 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) Mes 1 Mes 2 Fig. 8. von Mises stress results for te tool jig problem. Radius and radius B are defined in Fig. 2, and Mes 1, Mes 2 and are sown in Fig. 7. Te figures on te left sow te von Mises stress along radius, wereas te figures on te rigt sow te von Mises stress along radius B. It is interesting to note tat, for te problems considered in Figs. 4 and 7, te percentage improvement in stresses increases as te mes is refined. Naturally, te improvement is most important in te regions of ig stress gradients, wic, of course, is due to te fact tat te stresses s ðmþ are constant for te 4-node tetraedral finite element. In tese problems, we ave set te Poisson ratio to zero, to ensure consistency wit Ref. [1]; owever, te same level of improvement is also observed for non-zero values of Poisson ratio, for example, wen t ¼ 0:3, as long as te material is not almost or fully incompressible. Wen te medium is incompressible, as well-known, te 4-node displacement-based tetraedral element is not effective because it does not satisfy te inf sup condition [2,15,16] and is better not used. 4. Concluding remarks In tis paper we applied te approac given in Ref. [1] to establis a procedure for te calculation of improved stresses for te widely-used 4-node tree-dimensional displacement-based tetraedral element. s expected, wen we applied te procedure, we ave indeed seen a significant improvement in te stress predictions for all problems solved.

9 D.J. Payen, K.J. Bate / Computers and Structures 89 (2011) Tese results are quite encouraging, and te simple algoritm migt well be attractive in practice (after furter studies, see below) especially, for complex problems tat are expensive to analyse since te procedure allows te analyst to enance te stress predictions in localised regions of concern witout aving to refine te mes or re-analyse te model. Regarding future researc on te NPF-metod for stress predictions, as we pointed out already in Ref. [1], a strong matematical basis of te procedure would be of great value in order to identify te optimal stress assumptions and associated stress calculation domains to use. Given a specific sceme, teoretical studies of convergence and numerical studies on more complex problems are clearly needed to identify ow, and ow well, te predictions converge to te solutions sougt. Ten improvements to te algoritms used in Ref. [1] and erein may well be identified, and limitations may be establised. In addition, te use of te NPFmetod for stress calculation migt be explored in te analysis of sells, dynamic analyses, and te solution of nonlinear problems. References [1] Payen DJ, Bate KJ. Te use of nodal point forces to improve element stresses. Comput Struct 2011;89: [2] Bate KJ. Finite element procedures. Prentice Hall; [3] Capelle D, Bate KJ. Te finite element analysis of sells fundamentals. second ed. Springer; [4] Bate KJ, Dvorkin EN. formulation of general sell elements - te use of mixed interpolation of tensorial components. Int J Numer Met Eng 1986;22: [5] Bate KJ, Iosilevic, Capelle D. n evaluation of te MITC sell elements. Comput. Struct. 2000;75:1 30. [6] Bate KJ, Lee PS. Measuring te convergence beavior of sell analysis scemes. Comput Struct 2011;89: [7] Bucalem ML, Bate KJ. Te mecanics of solids and structures ierarcical modeling and te finite element solution. Springer; [8] Sussman T, Bate KJ. Studies of finite element procedures stress band plots and te evaluation of finite element meses. Eng Comput 1986;3: [9] Bate KJ, Bouzinov P. On te constraint function metod for contact problems. Comput Struct 1997;64: [10] Pantuso D, Bate KJ, Bouzinov P. finite element procedure for te analysis of termo-mecanical solids in contact. Comput Struct 2000;75: [11] rgyris JH. Recent advances in matrix metods of structural analysis. Progress in aeronautical sciences, vol. 4. Oxford: Pergamon Press; [12] Onimus S, Stein E, Walorn E. Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. Int J Numer Met Eng 2001;52: [13] Wriggers P, Reese S. note on enanced strain metods for large deformations. Comput Met ppl Mec Eng 1996;135: [14] Pantuso D, Bate KJ. On te stability of mixed finite elements in large strain analysis of incompressible solids. Finite Elements nal Des 1997;28: [15] Bate KJ. Te inf-sup condition and its evaluation for mixed finite element metods. Comput Struct 2001;79: [16] Capelle D, Bate KJ. On te ellipticity condition for model-parameter dependent mixed formulations. Comput Struct 2010;88:581 7.