Meshless EPM Solver Technology for Structural Mechanics Problems

Transcription

1 A technical note on Meshless EPM Solver Technology for Structural Mechanics Problems by The Computational Mechanic Company, Inc. (COMCO) 7800 Shoal Creek Blvd., Suite 290E Austin, TX and Altair Engineering 1757 Maplelawn, Troy, MI June 10, 1999 Executive Overview The Computational Mechanics Company (COMCO) is currently developing a new fast and robust solver for structural mechanics problems. It is based on a new solver technology called the Element Partition Method (EPM). This technology combines the modeling advantages and flexibility of finite element methods with the ease of use of the so-called meshless methods, and with computational performance compatible or faster than present finite element solvers. In the context of the design process, this new technology will allow analysts to solve the basic class of linear structural analysis problems in a "meshless" way, wherein the model discretization and solution (to a predefined accuracy level) is performed without user intervention. At the same time, in distinction from other classes of meshless methods, EPM can handle physics and model complexity similar to FEA and can, in fact, be combined with finite element entities. This technical note presents basic information about this technology and example s of its application to selected structural problems. A subsidiary of Altair Engineering. 1

2 Table of Contents 1. INTRODUCTION COMPETITIVE BENEFITS OF THE EPM TECHNOLOGY POTENTIAL BENEFITS OF EPM TO THE DESIGN PROCESS TECHNICAL OUTLINE OF EPM...5 Basic Technical Concept of EPM... 5 Technical Advantages of EPM NUMERICAL EXAMPLES...8 EPM Solution of a Typical Structural Problem... 8 EPM Performance On Badly Shaped Elements EPM Performance On Inconsistent Meshes Special Solution Functions in EPM Dynamic Analysis Introductory Examples of Applications of EPM to Shells Shell Shear Locking Behavior Shell Convergence Results CONCLUDING REMARKS REFERENCES...19 A subsidiary of Altair Engineering. 2

3 1. Introduction The Computational Mechanics Company (COMCO) is currently developing a new fast and robust solver for structural mechanics problems. It is based on a new solver technology, which promises to combine the modeling advantages and flexibility of Finite Element methods with the ease of use of so-called meshless methods [1, 2, 4], and with computational performanc ecompatible or faster than modern finite element codes. The solver is based on proprietary extensions of Meshless Cloud Methods [2, 4] and Generalized Finite Element methods [5] developed recently at COMCO and at the University of Texas. It is being implemented in an hp-adaptive context within the computational environment PHLEX [3]. Formally, the method is named Generalized Finite Element Partition of Unity Method or, for short, an EPM method. This technical note presents basic information about this technology and examples of its application to a selected set of structural problems. Due to the proprietary nature of this technology, we request that this note be treated as confidential. 2. Competitive Benefits of the EPM Technology The present state-of-the-art in mechanical simulation is dominated by the Finite Element method. While very effective in solution of a broad class of problems, this method requires a mesh of appropriate quality and element density. Typically, meshing is the most time-consuming part of the analysis process. Even with the present automated tetrameshers, many geometries cannot be easily meshed and require considerable effort from the analyst. Moreover, these meshes often contain elements of non-optimum shapes and sizes, which produce inaccurate solutions. Recently a class of so-called meshless methods has emerged as a possible remedy for this problem. This includes a variety of methods, such as Element-Free Galerkin, hp-cloud Methods [1, 2, 4], or external approximation technique used in the Procision software. 1 These techniques, while reducing in some way the model preparation time, have certain disadvantages, which hamper their application to a general class of problems. In particular: 1. Element-free Galerkin or meshless Cloud Methods greatly reduce meshing requirements as compared with finite element techniques. While they still require an underlying set of integration cells, the quality and consistency requirements for these cells are much less stringent than for finite element meshes. On the downside, these methods are generally much slower than the finite element method. Furthermore, they are not readily applicable to problems with mixed dimensionality (solids, shells and beams in a single model). 2. The external approximation technique used in the Procision software allows for very fast solution of three-dimensional problems without finite element mesh generation. However, the user is required to subdivide the domain into a number of sub-parts of rather simple shape. This is not easy to do for more complex domains. Moreover, the basic characteristics of this method essentially prevent it from being extendible to problems with multiple materials, complex nonlinear problems and, particularly, problems with mixed dimensionality (solids, shells and beams in a single model). 1 Procision is a trademark of Procision Analysis, Inc. A subsidiary of Altair Engineering. 3

4 The new EPM method overcomes the limitations of these techniques, while still providing the user with the benefits of easy model generation and fast solution with controllable accuracy. The main advantages of this technique can be summarized as follows: it simplifies model preparation by imposing minimum requirements on the consistency and quality of integration cells. This makes it possible to generate them in a hidden, automatic way, thus providing the user with meshless analysis, its performance is compatible or faster than that of the finite element method (the actual ratio depends on the characteristics of the model), it allows for use of special solution functions for inexpensive and accurate resolution of specific model features, such as crack tips, spot welds, holes, etc. it can handle shape and physics complexity similar to finite elements multiple materials, nonlinear problems, combinations of solids, shells and beams, MPC's, etc. it supports control of solution accuracy through error estimation and hp-adaptivity. Shortly, the EPM method combines and extends the best features of the finite element method while allowing for easier model preparation expected from the so-called meshless methods. 3. Potential Benefits of EPM to the Design Process For the end user, EPM can greatly shorten the analysis portion of the product development cycle. Moreover, it can allow for basic types of stress and vibration analysis to be performed by designers with minimum computational background, rather than require an FEA analyst. A general overview of a simplified design-analysis process utilizing EPM is shown in Figure 1. Generally, the user interacts with a CAD/CAE system, where he defines the geometry, material and loads for the given problem. In this setup, two general analysis scenarios are possible: Meshless Analysis : if the geometry is clean and contains well-defined entities (currently solids, later also shell surfaces) the user can request meshless analysis, which will automatically perform the following steps: 1. Auto-mesh the model to provide integration cells for EPM. Due to very relaxed requirements on the element size, aspect ratio and consistency, it is virtually guaranteed that the domain can be covered with integration cells using present automeshers. 2. Solve the problem using EPM, with error estimation and adaptivity for maximum efficiency. 3. Deliver the results to the CAE system or a specialized postprocessor. Simplified Analysis with Meshing: if the geometry is not clean or includes complex combinations of solids, shells, beams, etc., a fully encapsulated meshless analysis may not be possible. However, the user can still benefit from the EPM solver by: being able to mesh separate subdomains without requiring mesh consistency on the interfaces, generating a very fine mesh without expecting excessively high solution times, using special functions to resolve specific details rather than hand-generate a fine mesh, and being able to combine EPM with traditional finite element entities (beams, shells, rigids, MPC's, etc.). A subsidiary of Altair Engineering. 4

5 Meshless Analysis Simplified Analysis for Fragmented Geometries or Complex Models CAD/CAE system Ideas, ProEngineer, HyperMesh... Clean Geometry, Loads Hidden meshing (relaxed quality requirements) EPM solution -fast -accurate (error est.) Results Geometry, Loads Geometry Cleanup Meshing (relaxed quality and consistency requirements) EPM/FEA solution -fast -accurate (error est.) -special functions Results Figure 1. General scenarios for design analysis using EPM. 4. Technical Outline of EPM The EPM technology is a culmination of COMCO s research and development in the areas of adaptive finite element methods and meshless methods. While it requires an initial set of integration cells, similar to a traditional Finite Element mesh, it imposes significantly fewer restrictions during their generation. In particular, it is not sensitive to the aspect ratio of the elements and can handle localized mesh inconsistencies without degrading the solution quality. As a result of these relaxed requirements, it can dramatically improve the success ratio of automated mesh generation, allowing one to solve a very broad class of problems without human intervention. Moreover, the possibility of using customized shape functions for specialized solution features, such as spot welds, crack tips, etc. opens new avenues for easy and accurate modeling of problems where such features are present. Basic Technical Concept of EPM The EPM solver is based on proprietary extensions of certain versions of Partition of Unity Methods, namely Meshless Cloud Methods [2, 4] and Generalized Finite Element methods [5] developed recently at COMCO and at the University of Texas. The class of Partition of Unity Methods is based A subsidiary of Altair Engineering. 5

6 on the following approximation (for simplicity, the presentation below is limited to a one-dimensional setting these ideas extend naturally to two- and three-dimensional problems [1, 2, 5]): Let u(x) be a function defined on a domain Ω in R. Let the domain Ω be covered with a covering of N clouds ω α with centers at x α. Then, define a so-called partition of unity functions ϕ α (x) defined on clouds ω α and having the following property: ϕ α α C s 0 ( ωα ), ϕ ( x) = 1 α s 0, x Ω 1 α N (1) Examples of correct partitions of unity are Lagrangian finite elements (see Figure 2), moving least squares, Shepard functions, and others. Figure 2. One-dimensional finite element partition of unity. Having defined the partition of unity, we construct local approximations u α of u(x), such that u α is defined on ω α and u α is a good approximation of u(x). Then it can be shown that proper shape functions can be constructed as a tensor product of partitions of unity ϕ α with local approximations u α. In particular, for an element with nodes x α and x α+1 the shape functions can be constructed as: S α = α { ϕα, ϕα+ 1 } {1, uα, uα+ 1} = { ϕα, ϕα+ 1, ϕαuα, ϕα uα + 1, ϕα+ 1uα, ϕα+ 1u + 1} (2) Using the above shape functions, the usual Galerkin approximation can be defined for the function u(x) similarly as in classical finite element methods. However, the Partition of Unity methods have some unique advantages as compared with the finite element methods: 1. The finite element method can be recovered as a special case of Partition of Unity methods (by setting ϕ α (x) to be the standard shape functions and u α =1). 2. The local approximations u α can be chosen from a broad class of functions (as opposed to polynomial approximations used in finite elements) thus permitting developers to implement special functions fitted to the specific problem being solved. 3. The partition of unity ϕ α (x) can be defined using Lagrangian shape functions or any other method satifying the condition (1), thus opening way to various versions of so-called meshless methods. The Element Partition Method (EPM) is formally defined as a version of the Partition of Unity methods where ϕ α is built using Lagrangian linear shape functions, and the local approximations u α are constructed using polynomials or special functions representing specific features of the solution A subsidiary of Altair Engineering. 6

7 (such as crack tips, holes, connectors, etc.). COMCO has developed proprietary extensions of this method, which allow for effectively handling localized mesh inconsistencies. Technical Advantages of EPM EPM possesses several characteristics that set it apart from the classical Finite Elements and other meshless-type methods. In the computational model, the EPM methodology can handle: mixed meshes with tets, hexes, prisms, and arbitrarily collapsed hexes, elements with bad aspect ratios and large angles, and localized mesh inconsistencies (e.g. hex facing two tets or prisms). Under such relaxed requirements, and for correctly defined geometries, today s automated mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. This has been confirmed by introductory studies performed using Altair s mesh generator. In addition to such flexibility in model preparation, this technique has the following advantages as compared with the present state of-the-art solvers: computational performance compatible or faster than finite element methods (some additional speedup techniques are currently under development), a rigorous mathematical foundation with error estimation and adaptivity, providing reliable results and solution quality estimates, can be applied to solid models as well as to shell/plate models, can efficiently solve typical stress concentration problems without mesh refinements, and can incorporate specialized modeling approximations for specific model features, such as cracks, holes, fillets, reentrant corners, spot welds, etc. The underlying approximation technology can be understood as an extension of traditional finite element methods, and it does not impose any additional limitations. Therefore, in contrast to some recent techniques based on meshless concepts, EPM can be applied to: domains with multiple materials, models which incorporate traditional discrete finite elements (beams, rods, plates, shells, etc.) including superelement-based substructures, the solution of nonlinear problems (large deformation and/or nonlinear material models), and optimization techniques, including shape and topology optimization. In short, this new technology combines the best features of finite element methods with the ease of problem formulation expected from meshless techniques, accuracy control through error estimation and with superior computational efficiency. A subsidiary of Altair Engineering. 7

8 5. Numerical Examples The base version of the EPM technology has been implemented into COMCO s PHLEX hp-adaptive computational kernel [3]. As an illustration, we present a few selected solutions, highlighting the overall benefits of this technology and its specific advantages as compared with the finite element method. In particular, the examples below include: 1) EPM solution of a typical structural problem, 2) illustration of EPM performance on bad elements, 3) illustration of EPM performance on mismatched meshes, 4) an illustration of special solution functions in EPM, 5) an illustration of dynamic analysis with cracks, and 6) an illustration of application to shell analysis. EPM Solution of a Typical Structural Problem In this section we present an introductory solution of a typical bracket problem. The model was solved using both finite elements and EPM method. A summary comparison of the results discussed below is shown in Table 1. The finite element mesh for this model and the boundary conditions are shown in Figure 3 it has 15,527 tetrahedral elements and 3,849 vertex nodal points. The material is assumed to be linearly elastic with Young's modulus E= and Poisson's ratio ν = The component is fixed at both supports and there is a uniformly distributed load T = [-1 0 0] applied at the upper hole. The problem was solved by a finite element method using quadratic Tet 10 elements with 76,797 degrees of freedom (equations). The results (von Mises stress distribution) are shown in Figure 4. Figure 3. Bracket geometry and boundary conditions. A subsidiary of Altair Engineering. 8

9 Figure 4. Finite element solution: Von Mises stress distribution. This problem was also solved using the EPM method, with approximation points corresponding to the vertices of the finite element mesh, and with the order of approximation equal to p=2. The EPM covering based on the finite element mesh is shown in Figure 5 it contains 46,188 degrees of freedom. The resulting stress distribution is shown in Figure 6. It can be observed that the overall solution is similar to that of the finite element method and indeed the quality of the stress distribution is superior, in spite of the lower number of degrees of freedom (in both cases raw stresses are shown without any smoothing). Figure 5. An automatically generated EPM covering based on an original finite element mesh. A subsidiary of Altair Engineering. 9

10 Figure 6. Von Mises stress distribution calculated with EPM. A brief comparison of the finite element results with the EPM method is shown in the table below. While the method is still under development and these results are tentative, it is clear that EPM produces high-quality results at a fraction of the time required by the quadratic finite elements, while additionally providing the benefits of easy model generation typical of meshless methods. Method: FE (Tet 10) EPM (p=2) Number of equations 76,797 46,188 CPU time (min.) Strain Energy von Mises stress at (-85, 50, 5) Table 1. A comparison of results between the finite element method and EPM. It should also be noted that, while the above example was solved using a correct finite element mesh, the EPM is capable of handling a much less regular covering of the domain. Relevant examples are shown below. EPM Performance On Badly Shaped Elements One of the advantages of the EPM method is that it can provide high-quality results even on badly shaped elements. This is primarily because the elements are merely used as integrations cells and A subsidiary of Altair Engineering. 10

11 support for the partition of unity (POU), while the polynomial approximation is not affected by the element shape (like it is the case for isoparametric finite elements). As an illustration of this advantage, a simple cantilever beam was solved using both finite element and EPM methods. The finite element mesh for this problem is shown in Figure 7 the same mesh was used to define integration cells and a partition of unity for EPM. In both cases, a quadratic order of approximation was used. Clearly, the mesh has some very elongated tetrahedra, which degrade the quality of finite element solution. A comparison of finite element solution with EPM is presented in Table 2. Since the maximum stress for this problem is not well defined due to presence of singularities, the maximum displacement at the tip of the beam and the total strain energy were used for comparison. Additionally, the table shows the value of stress σ xx at the top surface in the center of the beam length (away from load and support singularities). It can be observed that, while the finite element solution, even with a third order approximation, is too stiff, the EPM method provides high quality results. It is worth noting that finite element stress calculation within the slender elements is completely off. It is somewhat better in the zone near the fixed end, where the elements have acceptable shapes. Still, even in that zone the stresses from EPM are more accurate than FEA. Figure 7. Cantilever beam problem - problem definition and a tetrahedral mesh with very elongated elements. Method: FEA (p=3, 522 eqn) EPM (p=3, 720 eqn) Exact (beam theory) Max. displ Strain energy σ xx at (12.5,0.5,1) Table 2. A comparison of solution obtained on very slender elements using finite element method and EPM. A subsidiary of Altair Engineering. 11

12 EPM Performance On Inconsistent Meshes One of valuable potential advantages of the EPM method is that it can be effectively used on meshes with inconsistent elements, where the element nodes do not match along certain surfaces. A simple, yet representative example of this situation is shown in Figure 8. It is a T-shaped domain subjected to a bending load, with mesh inconsistencies present along three surfaces (note that the mesh on the crossbar does not even place nodes on the inner T-corner). The EPM results (von Mises stress distribution) are shown in Figure 9. The solution satisfies all the necessary continuity and consistency requirements. Moreover, it captures the stress concentration on the inner T-corner (a slight shift of the concentration to the nearest active node can be noticed). Figure 8. T-shaped domain under bending load: problem statement and inconsistent tetrahedral mesh. Figure 9. Von Mises stress distribution for the T-shaped domain. Another example of EPM performance on inconsistent meshes that include very slender tetrahedral elements is a three-dimensional plate with a hole, shown in Figure 10. The mesh includes both hexahedral and tetrahedral elements, mismatched along a relatively large section of the domain. In A subsidiary of Altair Engineering. 12

13 spite of this discrepancy, the displacements and stresses calculated with EPM (see Figure 11 and Figure 12) exhibit very good symmetry. Also, the EPM results compare favorably with an analytical solution: the calculated maximum longitudinal stress component σ xx was equal 3.44, while the analytical value for two-dimensional problem is Figure 10. Inconsistent initial mesh for a three-dimensional plate tension problem (top view). The brick elements are enriched to p=2 and tetrahedral elements to p=4 in the in-plane direction, with p=1 in the out-of-plane direction. Figure 11. X-displacements calculated on inconsistent mesh. Figure 12. Von Mises stress calculated on inconsistent mesh. Special Solution Functions in EPM One of the most unique features of the EPM method is the possibility of incorporating special solution functions, which resolve specific characteristics of the problem much better than polynomial shape functions used in the finite element method. The most typical cases where such functions can be beneficial involve stress singularities and sharp stress concentrations. Some examples include: crack tips, holes of various shapes, A subsidiary of Altair Engineering. 13

14 small geometrical features, such as fillets and chamfers, and special connections, such as bolts or spot welds. While in the Finite Element method resolution of such features requires very fine local meshes, implementation of special functions in EPM can greatly reduce the problem size and computational cost, while improving the accuracy. Presently, the EPM solver incorporates special functions developed for crack tips, which are characterized by a strong stress singularity of the type 1/vr, where r is a distance from the crack tip. As an example of effectiveness of these special functions, we present a problem of a transient load applied to a cracked plate (this problem has an analytical solution). The overall problem statement is shown in Figure 13. The example was solved using both the EPM and finite element methods, with comparable level of accuracy expected for stress intensity factors from both methods. Figure 13. Problem statement for a cracked plate under dynamic loading. A comparison of EPM discretization with a refined finite element mesh in the vicinity of the crack tip is shown in Figure 14. While the finite element mesh has been strongly refined around the crack tip, the EPM method incorporates a special function associated with the singularity (note that in EPM the crack actually cuts across the elements). The transient solution for stress intensity factor K I is shown and compared with analytical results in Figure 15. Clearly, both finite element and EPM solutions reached similar levels of accuracy, yet the finite element solution required aligning the elements with the crack and a strongly refined mesh at the crack tip. A subsidiary of Altair Engineering. 14

15 Figure 14. Discretization at the crack tip: a) EPM with singular functions, and b) adapted FEA. Figure 15. History of K I stress intensity factor at the crack tip. Dynamic Analysis In contrast to some other types of meshless methods, EPM can be applied to a broad class of physical problems, similar in scope to that of FEA. This includes, in particular, dynamics, nonlinear problems (both geometric and material nonlinearities), buckling, CFD, flow-structure interaction, etc. As an illustration of a transient dynamic application, we present an impact analysis of a yacht hull with pre-existing cracks. The problem statement is shown in Figure 16a where the hull is modeled as a thin solid discretized with EPM, and the impact force is applied to a bow section for a duration of 10-5 sec. A snapshot of the deformation of the hull at a selected time step, colored by the displacement A subsidiary of Altair Engineering. 15

16 magnitude, is shown in Figure 16b and a plot of the Von Mises stress is shown Figure 17 (note that the cracks cut across the interiors of EPM integration cells). Figure 16. Problem statement a) and displacement magnitude b) for a hull under impact load. Figure 17. Von Mises stress on a cracked hull under impact load (displacement scale exaggerated). Introductory Examples of Applications to Shells As it was mentioned previously, to-date the main thrust of EPM development has been focused on three-dimensional problems. However, some basic examples of application of Partition of Unity Methods, of which EPM is a member, to solution of plate problems have been performed before and we present here a representative example. In particular, some Mindlin's plate problems are solved using the hp-cloud method [1, 2]. The only difference between hp clouds and EPM is in the choice of the partition of unity. Therefore, all the results presented here can also be obtained using another partition of unity, as the one used in the EPM. A subsidiary of Altair Engineering. 16

17 Shell Shear Locking Behavior Shear locking occurs when the approximation functions are unable to meet the requirements for allowing null transversal shear deformations as the plate becomes thin. One effective approach used in the FEM to overcome this difficulty is the use of finite elements of degree p=3 or higher (depending on the relative thickness of the plate). The same approach, however, cannot be used in meshless methods based on moving least square (MLS) functions, such as the Diffuse Element Method, Element Free Galerkin Method and Reproducing Kernel Particle Methods. This happens because the cost of building a MLS approximation that can reproduce polynomials of degree greater or equal to 3 is prohibitively high in those methods. In addition to that, the support of these functions, i.e., the region where they are non-zero, is considerably larger than finite element shape functions of the same polynomial degree. As a consequence, the bandwidth of the stiffness matrix would also be much bigger than in the FEM for the same degree of approximation, and the handling of nodes at or near the boundary would require special procedures to impose boundary conditions. In contrast, the EPM/hp-Cloud framework allows for construction of high order p approximations without the inversion of any matrix. Furthermore, the dimension of the support of the hp-cloud functions does not have to increase with the degree p of these functions (as in the case of MLS functions). These properties of the hp-cloud or EPM functions make them very appealing candidates to solve Mindlin's plate problem in a meshless framework. This first example consists of a simply supported plate subjected to a uniformly distributed load. Only one forth of the plate is discretized and appropriate symmetry boundary conditions are applied. The domain is discretized using a uniform distribution of 5x5 hp-cloud nodes as shown in Figure 18. Approximations with degree p=3 and p=4 are used. Figure 18. Plate model and discretization with 5x5 nodes. A subsidiary of Altair Engineering. 17

18 Figure 19. Normalized transversal displacement at the center of the plate. The normalized transversal displacement at the center of the plate is shown on Figure 19. The results are normalized with respect to the Kirchoff-Love thin plate solution. It can be observed that as the plate becomes thin both discretizations converge to the thin plate solution as expected. Shear locking appears for the cubic discretization only when the relation L/t is larger than 10,000. No shear locking is observed in the case of the quartic discretization in the range of L/t computed (up to L/t = 200,000). These results confirm that higher-order approximation in meshless methods based on partition of unity releases the locking phenomenon in very thin shell models similarly as the high-quality p-shell finite element models. Shell Convergence Results In this example, the convergence of the hp-cloud plate solution in the case of h and p refinements is investigated. The plate has all its edges simply supported and is subjected to a uniformly distributed load. The thickness is t = 0.1 mm and the plate has dimensions of 16x16 mm. Uniform discretizations with 5, 25 and 87 nodes and polynomial degrees 0 <= p <= 4 are used. The analytic solution for this problem is known. The results are shown in Figure 20. The relative error is measured in the L 2 norm. It can be observed that the rate of convergence increases with p-enrichment. This is typical of spectral methods which converge exponentially with p enrichment in the case of smooth solutions. Note, for example, that in the case of the discretization with 25 nodes, the discretization error decreases several orders of magnitude when the approximation is enriched from p=2 to p=3. A subsidiary of Altair Engineering. 18

19 Figure 20. Convergence analysis of the hp-cloud plate models. The relative error of the hp-cloud solution is measured in the L 2 norm. 6. Concluding Remarks This technical note summarized the benefits and future potential of an new version of so-called meshless methods developed at COMCO. This new method promises to combine the modeling advantages and flexibility of finite element methods with the ease of use of meshless methods, and with computational performance compatible or faster than modern finite element codes (up to an order of magnitude faster with proprietary techniques presently under development). It should be noted that many aspects of the method are still under development and its full potential has not yet been explored. It can be expected that with additional speedup techniques, specialized integration schemes and with application of special functions for modeling of typical geometrical features (spot welds, holes, fillets, cracks, etc.) the benefits of this method will go beyond the introductory examples shown above. 7. References 1. Liszka, T.J., Duarte C.A.M. and Tworzydlo, W.W., "hp-meshless cloud method'', Comp. Meths. Appl. Mech. Engng, 139, pp , Duarte, C. A. M. and Oden, J. T., An hp Adaptive Method Using Clouds, Comp. Meth. Appl. Mech. Eng, 139, , A subsidiary of Altair Engineering. 19

20 3. Liszka, T.J., Tworzydlo, W.W., Bass, J.M., Sharma S.K., Westermann T.A. and Yavari B.B., "PHLEX an hp-adaptive finite element kernel for solving coupled systems of partial differential equations in computational mechanics, Comp. Meths. Appl. Mech. Engng, 150, pp , Oden, J. T., Duarte, C. A. and Zienkiewicz, O. C., A New Cloud-Based hp Finite Element Method, Comp. Meth. Appl. Mech. Eng, 153, pp , Duarte, C. A., Babuska, I. and Oden, J. T., Generalized Finite Element Methods for Three Dimensional Structural Mechanics Problems, International Conference on Computational Engineering Science, Atlanta, GA, October 5-9, A subsidiary of Altair Engineering. 20