The proper generalized decomposition for advanced numerical simulations ch28

The proper generalized decomposition for advanced numerical simulations ch28 Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom. Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab© software used to generate these examples.

28 Templates and Morphing 28–1 28–2 Chapter 28: TEMPLATES AND MORPHING TABLE OF CONTENTS Page §28.1 FINITE ELEMENT TEMPLATES §28.1.1 The Fundamental Decomposition §28.1.2 Constructing the Component Stiffness Matrices §28.1.3 Basic Stiffness Properties §28.1.4 Constructing Optimal Elements 28–3 28–3 28–4 28–4 28–5 §28.2 FROM 4-NODE RECTANGLE TO BEAM 28–6 §28.2.1 The Basic Stiffness 28–6 §28.2.2 The Higher Order Stiffness 28–8 §28.2.3 Constructing the Higher Order Stiffness 28–11 §28.3 MORPHING TO BEAM 28–12 28–2 28–3 §28.1 FINITE ELEMENT TEMPLATES This Chapter provides an introduction to two new concepts in the development of high-performance finite elements: templates and morphing A finite element template, or template is an algebraic form for element matrices, which contains free parameters Setting those parameters to specific values produces element instances The template is constructed by the process of direct fabrication described in Chapter 23 The transformation of a finite element or macroelement into a simpler model through constraints will be called element morphing The technique has received cyclic attention in the FEM literature Historically the construction of thick shell elements by the degenerate 3D solid approach represents one of the most important examples, and one that produced significant theretical advances during the 1970s In fact the majority of the applications of this technique involves the construction of bending elements from elasticity elements In the present Chapter the technique is illustrated by the example of morphing plane stress elements to beams §28.1 FINITE ELEMENT TEMPLATES A finite element template, or simply template, is an algebraic form that represents element-level stiffness equations, and which fulfills the following conditions: (C) Consistency: the Individual Element Test (IET) form of the patch test, introduced by Bergan and Hanssen is passed for any element geometry (S) Stability: the stiffness matrix satisfies correct rank and nonnegativity conditions (P) Parametrization: the element stiffness equations contain free parameters (I) Invariance: the element equations are observer invariant In particular, they are independent of node numbering and choice of reference systems The first two conditions: (C) and (S), are imposed to ensure convergence Property (P) permits performance optimization as well as tuning elements to specific needs Property (I) helps predictability and benchmark testing Setting the free parameters to numeric values yields specific element instances §28.1.1 The Fundamental Decomposition A stiffness matrix derived through the template approach has the fundamental decomposition K = Kb (αi ) + Kh (β j ) (28.1) Here Kb and Kh are the basic and higher-order stiffness matrices, respectively The basic stiffness matrix Kb is constructed for consistency and mixability, whereas the higher order stiffness Kh is constructed for stability (meaning rank sufficiency and nonnegativity) and accuracy As further discussed below, the higher order stiffness Kh must be orthogonal to all rigid-body and constantstrain (curvature) modes In general both matrices contain free parameters The number of parameters αi in the basic stiffness is typically small for simple elements For example, in the 3-node, 9-DOF KPT elements considered here there is only one basic parameter, called α This number must be the same for all elements in a mesh to insure satisfaction of the IET 28–3 28–4 Chapter 28: TEMPLATES AND MORPHING w1 z,w EI = constant w2 θ1 L θ2 One free parameter 0 0   EI   −1  + β E I K = Kb + K h =  L L3 0 0  −1 x  −2L  −4 −2L −2L L2 2L L2 −4 2L 2L −2L  L2   2L L2 Figure 28.1 Template for Bernoulli-Euler prismatic plane beam On the other hand, the number of higher order parameters β j can be in principle infinite if certain components of Kh can be represented as a polynomial series of element geometrical invariants In practice, however, such series are truncated, leading to a finite number of β j parameters Although the β j may vary from element to element without impairing convergence, often the same parameters are retained for all elements As an illustration Figure 28.1 displays the template of a simple one-dimensional element: a 2-node, 4-DOF plane Bernoulli-Euler prismatic beam This has only one free parameter: β, which scales the higher order stiffness A simple calculation shows that its optimal value is β = 3, which yields the well-known Hermitian beam stiffness This is known as a universal template since it include all possible beam elements that satisfy the foregoing conditions §28.1.2 Constructing the Component Stiffness Matrices The basic stiffness that satisfies condition (C) is the same for any formulation It is simply a constant stress hybrid element For a specific element and freedom configuration, Kb can be constructed once and for all The formulation of the higher order stiffness Kh is not so clear-cut, as can be expected because of the larger number of free parameters It can be done by a variety of techniques, which are summarized in a article by Felippa, Haugen and Militello cited in Chapter 23 Of these, one has proven exceedingly useful for the construction of templates: the ANDES formulation ANDES stands for Assumed Natural DEviatoric Strains It is based on assuming natural strains for the high order stiffness For plate bending (as well as beams and shells) natural curvatures take the place of strains Second in usefulness is the Assumed Natural DEviatoric STRESSes or ANDESTRESS formulation, which for bending elements reduces to assuming deviatoric moments This technique, which leads to stiffness templates that contain inverses of natural flexibilities, is not considered here 28–4 28–5 §28.1 FINITE ELEMENT TEMPLATES §28.1.3 Basic Stiffness Properties The following properties of the template stiffness equations are collected here for further use They are discussed in more detail in the article by Felippa, Haugen and Militello cited in Chapter 23 Consider a test displacement field, which for thin plate bending will be a continuous transverse displacement mode w(x, y) [In practical computations this will be a polynomial in x and y.] Evaluate this at the nodes to form the element node displacements u These can be decomposed into u = ub + uh = ur + uc + uh , (28.2) where ur , uc and uh are rigid body, constant strain and higher order components, respectively, of u The first two are collectively identified as the basic component ub The matrices (28.1) must satisfy the stiffness orthogonality conditions Kb ur = 0, Kh ur = 0, K h uc = (28.3) while Kb represents exactly the response to uc The strain energy taken up by the element under application of u is U = 12 uT Ku Decomposing K and u as per (28.1) and (28.2), respectively, and enforcing (28.3) yields U = 12 (ub + uh )T Kb (ub + uh ) + 12 uhT Kh uh = Ub + Uh (28.4) Ub and Uh are called the basic and higher order energy, respectively Let Uex be the exact energy taken up by the element as a continuum body subjected to the test displacement field The element energy ratios are defined as ρ= U = ρb + ρh , Uex ρb = Ub , Uex ρh = Uh Uex (28.5) Here ρb and ρh are called the basic and higher order energy ratios, respectively If uh = 0, ρ = ρb = because the element must respond exactly to any basic mode by construction For a general displacement mode in which uh does not vanish, ρb is a function of the αi whereas ρh is a function of the β j §28.1.4 Constructing Optimal Elements By making a template sufficiently general all published finite elements for a specific configuration can be generated This includes those derivable by orthodox techniques (for example, shape functions) and those that are not Furthermore, an infinite number of new elements arise The same question previously posed for PVPs arises: Can one select the free parameters to produce an optimal element? The answer is not yet known for general elements The main unresolved difficulty is: which optimality conditions must be imposed at the local (element) level? While some of them are obvious, for example those requiring observer invariance, most of the others are not The problem is that a detailed connection between local and global optimality is not fully resolved by conventional FEM error analysis Such analysis can only provide convergence rates expressed as C h m in some error 28–5 28–6 Chapter 28: TEMPLATES AND MORPHING norm, where h is a characteristic mesh dimension and m is usually the same for all template instances The key to high performance is the coefficient C, but this is problem dependent Consequently, verification benchmarks are still inevitable As noted, conventional error analysis is of limited value because it only provides the exponent m, which is typically the same for all elements in a template It follows that several template optimization constraints discussed later are heuristic But even if the local-to-global connection were fully resolved, a second technical difficulty arises: the actual construction and optimization of templates poses formidable problems in symbolic matrix manipulation, because one has to carry along arbitrary geometries, materials and free parameters Until recently those manipulations were beyond the scope of computer algebra systems (CAS) for all but the simplest elements As personal computers and workstations gain in CPU speed and storage, it is gradually becoming possible to process two-dimensional elements for plane stress and plate bending Most three-dimensional and curved-shell elements, however, still lie beyond the power of present systems Practitioners of optimization are familiar with the dangers of excessive perfection A system tuned to operate optimally for a narrow set of conditions often degrades rapidly under deviation from such conditions E and h constant y x L H Figure 28.2 The example element: a 4-node, 8-dof rectangular plate in plane stress §28.2 FROM 4-NODE RECTANGLE TO BEAM The morphing technique can be illustrated in the morphing of a 4-node plane stress rectangular element to a beam element We begin by constructing the plane stiffness matrix using the Free Formulation or FF The element geometry is depicted in Figure 28.2 It has lengths L and H in the x and y directions, respectively, uniform thickness h and material properties The latter are represented by the × matrix E of elastic moduli that relate stresses to strains: σ = Ee Using the FF the rectangle stiffness is decomposed into two parts: K = Kb + K h (28.6) where Kb is the basic stiffness, which takes care of consistency, and Kh is the higher order stiffness, which takes care of accuracy and stability 28–6 28–7 §28.2 FROM 4-NODE RECTANGLE TO BEAM tx = t y = − 12 σ¯ x x H h σ¯ H h xx σ¯ x x tx = −σ¯ x x , t y = − 12 σ¯ x x H h tx = σ¯ x x , t y = σ¯ H h xx tx = t y = σ¯ Lh yy σ¯ Lh yy tx = 0, t y = σ¯ yy σ¯ yy tx = t y = tx = t y = node forces surface tractions stresses tx = 0, t y = −σ¯ yy − 12 σ¯ yy Lh − 12 σ¯ yy Lh τ¯ H h xy − 12 τ¯x y H h tx = 0, t y = τ¯x y τ¯ Lh xy τ¯ Lh xy τ¯x y tx = 0, t y = τ¯x y tx = 0, t y = −τ¯x y − 12 τ¯x y Lh tx = −τ¯x y , t y = − 12 τ¯x y Lh τ¯ H h xy − 12 τ¯x y H h Figure 28.3 “Lumping” of constant stress components into node forces through the surface tractions on element sides §28.2.1 The Basic Stiffness The basic stiffness is derived as that of a equilibrium-stress-assumed hybrid element, in which the assumed stresses are constant over the element Note that an constant-stress field automatically satisfies the stress equilibrium equations if the body forces vanish There is no need, however, to go explicitly through the flexibility matrix F and its inverse as in the general derivation of hybrid elements covered in Chapter 16, as long as the material properties are constant over the element 28–7 28–8 Chapter 28: TEMPLATES AND MORPHING For the plane stress element we therefore chose the following stress assumption within the element: σx x σ yy σx y σ¯ x x σ¯ yy σ¯ x y = (28.7) where σ¯ x x , σ¯ yy and σ¯ x y are constant stress values On the element boundaries those stresses produce constant tractions that will be denoted by t¯x and t¯y It is convenient for visualization purposes to separate the individual effects of each stress component, as illustrated in Figure 28.3 These surface tractions are then “lumped” at the corner nodes into node forces, as illustrated in that Figure The result of the force-lumping process can be expressed in matrix form as follows:   ¯  −H f x1 ¯ f   y1    ¯   H  f x2    ¯   f y2   h =  ¯    H  f x3    ¯    f y3    ¯  f x4 −H ¯ f y4 0 −L −L L L  −L −H   −L   σ¯ x x H   σ¯ yy L   σ¯ x y H   L −H (28.8) or ¯f = Lσ ¯ (28.9) Matrix L is called the force-lumping matrix Since the constitutive matrix is constant over the element, σ ¯ = E¯e It may be shown (using the Principle of Virtual Work) that ¯ e¯ = Bu, where B¯ = LT /(Ah) (28.10) Therefore ¯f = (1/Ah)LELT u = Kb u, from which the basic stiffness follows as Kb = LELT Ah (28.11) Equation (28.11) is the general expression of the basic stiffness for any element if Ah is replaced by the element volume V It is seen that it requires only the construction of the force-lumping matrix L, because E is data The process of constructing L is analogous to that of lumping a distributed force to nodes, hence its name For more complex elements the lumping cannot be done by statics alone, and assumptions on boundary motions come into play For the present element, Kb has rank of at most 3, since that is the maximum rank of E Consequently Kb is rank deficient and cannot provide a stable element by itself The addition of the higher order stiffness is needed to attain the proper rank of = − 28–8 28–9 §28.2 u x = q1 , u x = x q4 , uy = uy = FROM 4-NODE RECTANGLE TO BEAM u x = 0, u y = q2 u x = −y q3 , u y = xq3 u x = 0, u y = y q5 u x = 12 y q6 , u y = 12 x q6 Figure 28.4 The r c modes (rigid-body and constant-strain modes) for the FF of the 4-node plane stress rectangle §28.2.2 The Higher Order Stiffness The higher order stiffness Kh can be constructed by different methods Some are based on displacements, some on strains, some on stresses, some on a combination of these The FF is entirely based on displacement modes Unlike shape functions, modes are patterns of clear physical meaning that not have to satisfy interelement compatibility This results in increased freedom for the “element designer,” hence the name of the formulation For the standard FF one must assume as many displacement modes as the number of element degrees of freedom (DOF) That is, modes for the example element These modes must also be linearly independent Figures 28.4 and 28.5 show the assumed modes for the present development These are broken down into three sets: Rigid body modes, or r -modes For plane stress elements there are three independent modes of this type There can be conveniently selected as the translation along x, translation along y, and rotation about the x, y origin The three modes can be mathematically expressed as shown in Figure 28.3 The mode amplitudes are characterized by the generalized coordinates coefficients q1 , q2 , q3 Constant strain modes, or c-modes For plane stress there are three independent modes of this type As their name suggests, they are the displacement patterns that produce constant strain states ex x , e yy and ex y = 12 γx y over the element They are expressed as illustrated in Figure 28.3 The higher order modes, or h-modes In the FF, the number of displacement modes must be equal to the number of nodal-displacement degrees of freedom of the element The deficit of = − is filled with higher order modes (h-modes) For a rectangular plane stress element, the simplest choice of h-modes are shown in Figure 28.5 The nonconforming or incompatible modes correspond to pure bending whereas the conforming or compatible modes is associated with a antiplane shear motion Rather than committing to a particular set, one can parametrize the choice as Mode 7: u x = x yq7 , u y = − 12 x q7 χ, Mode 8: u x = − 12 y q8 χ, 28–9 u y = x yq8 , (28.12) 28–10 Chapter 28: TEMPLATES AND MORPHING Nonconforming (incompatible) set Conforming (compatible) set u x = x y q7 , u x = 0, u x = x y q7 , uy = u y = x y q8 u y = − 12 x q7 u x = − 12 y q8 , u y = x y q8 Figure 28.5 The h modes (higher order modes) for the FF of the 4-node plane stress rectangle where χ is a scalar parameter If χ = one obtains the conforming set, whereas if χ = one gets the nonconforming set Combining the foregoing mode assumptions into a single matrix equation gives  u= ux uy = −y x y xy − 12 χ y x y x − 12 χ x xy  q1  q2     q3     q4    = Nq q  q5     q6    q7 q8 (28.13) Each column of Nq correspond to an individual mode The first three columns pertain to r modes, the next three to c modes and the last two to h modes The x y strain components derived from this displacement assumption may be written e= ∂u x /∂ x ∂u y /∂ y ∂u x /∂ y + ∂u y /∂ x = 0 0 0 0 0 0 y (1 − χ)x x (1 − χ)y q = Bq q (28.14) From (28.14) one can explicitly separate the contribution or r , c and h modes as follows e = Br qr + Bc qc + Bh qh = Iqc + Bh qh , (28.15) Here Br is × (take columns through 3), Bc is × (take columns through 6) and Bh is × (take the last two columns, and 8) Note that Br = because rigid body modes produce 28–10 28–11 §28.2 FROM 4-NODE RECTANGLE TO BEAM no strains whereas Bc = I is the identity matrix, because the generalized coordinates q4 , q5 , q6 are identified with the strain components It is seen that the strain decomposes naturally into e = ec + e h (28.16) where ec are constant over the element and eh are called the higher order strains It is easy to show that for this particular element and modal assumptions h e d A = ec Ah = ec V, h eh d A = A A hBh qh d A = 0, Bh d A = because A A (28.17) Consequently ec is effectively the mean strain over the element, while the mean value of eh is zero If these conditions are verified for an element, ec is called the mean strain e¯ The vanishing of Bh over the element area receives the name energy orthogonality condition §28.2.3 Constructing the Higher Order Stiffness The FF higher order stiffness for a general element is Kh = HhT Sh Hh , Sh = V (28.18) BhT EBh d V, where Hh is the matrix that relates qh = Hh u, and Sb is a generalized stiffness in terms of the qh coordinates For the example problem, array qh has length because it contains components q7 and q8 Therefore Sh is × and Hh is × The expression of Sh with d V = f d : Sh = hBhT EBh d x d y = H/2 L/2 h H/2 −L/2 y 0 x (1 − χ)x E (1 − χ)y y (1 − χ)x x (1 − χ)y d x d y (28.19) For isotropic material characterized by elastic modulus E and Poisson’s ratio ν the generalized h stiffness is diagonal: Sh = Eh H L 12(1 − ν ) H + 12 (1 − χ)(1 − ν)L L + (1 − χ)(1 − ν)H (28.20) in which Sh retains the free parameter χ introduced in the definition of h modes The derivation of the Hh matrix for general FF elements involves the construction of the transformatiuon matrix u = Gq by evaluating the FF modes at the node points Matrix G must be square and nonsingular Invertion gives q = Hu, where H = G−1 Finally Hh is extracted by partitioning H appropriately by columns so that qh = Hh u This process may entail heavy algebra and is thus best done with symbolic-algebra manipulation programs It can be bypassed, however, for this particular element because of its simple geometry 28–11 28–12 Chapter 28: TEMPLATES AND MORPHING Plane beam morphing h H L Figure Morphing a rectangular plate mesh unit to beam From geometric inspection of Figure 28.5 one obtains q7 = (u x1 − u x2 + u x3 − u x4 )/(H L) and q8 = (u y1 − u y2 + u y3 − u y4 )/(H L) Consequently Hh = HL 0 −1 0 −1 0 −1 0 , −1 (28.21) and the derivation of Kh is complete This matrix may be scaled by an arbitrary positive scalar, which will be called β = − γ : Kh = β HhT Sh Hh , with β > (28.22) Note that this scaled Kh is function of two free parameters: β = − γ and χ Combining the basic and higher-order stiffness matrices we arrive at the two-parameter template: K = Kb + Kh (β, χ) = LELT + βHhT Sh (χ)Hh Ah (28.23) If β = and χ = one recovers the well known isoparametric element On the other hand, if β = χ = one obtains an element that is free from parasitic shear §28.3 MORPHING TO BEAM Next we study the mapping of the rectangular element to a beam-column with the six degrees of freedom defined in Figure 28.6 The degrees of freedom of the quad and the beam-column are 28–12 28–13 §28.3 MORPHING TO BEAM collected in array ub From inspection, the appropriate transformation equations are    u x1  u y1       u x2       u y2   uq =  =  u x3       u y3      u x4 u y4 H 0 0 0 0 0 − 12 H 0 1 0 The morphed stiffness is Kb = TT Kq T To be completed in a paper under preparation 28–13  0   0  u1  H   w1      θ1     = Tub − 12 H   u     w2  0 θ2 0 (28.24) ... for the “element designer,” hence the name of the formulation For the standard FF one must assume as many displacement modes as the number of element degrees of freedom (DOF) That is, modes for. .. modes) for the FF of the 4-node plane stress rectangle where χ is a scalar parameter If χ = one obtains the conforming set, whereas if χ = one gets the nonconforming set Combining the foregoing... §28.1.3 Basic Stiffness Properties The following properties of the template stiffness equations are collected here for further use They are discussed in more detail in the article by Felippa,