The proper generalized decomposition for advanced numerical simulations ch18

The proper generalized decomposition for advanced numerical simulations ch18 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.

18 Hexahedron Elements 18–1 18–2 Chapter 18: HEXAHEDRON ELEMENTS TABLE OF CONTENTS Page §18.1 INTRODUCTION §18.1.1 Natural Coordinates §18.1.2 Corner Numbering Rules §18.2 THE EIGHT NODE (TRILINEAR) HEXAHEDRON 18–3 18–3 18–3 18–4 §18.3 THE 20-NODE (SERENDIPITY) HEXAHEDRON 18–5 §18.4 THE 27-NODE HEXAHEDRON §18.5 PARTIAL DERIVATIVES §18.5.1 The Jacobian §18.5.2 Computing the Jacobian Matrix §18.6 THE STRAIN DISPLACEMENT MATRIX 18–6 18–7 18–7 18–7 18–8 §18.7 STIFFNESS MATRIX EVALUATION §18.7.1 Selecting the Integration Rule 18–9 18–9 EXERCISES 18–2 18–10 18–3 §18.1 INTRODUCTION §18.1 INTRODUCTION The generalization of a quadrilateral three-dimensions is a hexahedron, also known in the finite element literature as brick A hexahedron is topologically equivalent to a cube It has eight corners, twelve edges or sides, and six faces Finite elements with this geometry are extensively used in modeling three-dimensional solids Hexahedra also have been the motivating factor for the development of “Ahmad-Pawsey” shell elements through the use of the “degenerated solid” concept The construction of hexahedra shape functions and the computation of the stiffness matrix was greatly facilitated by three advances in finite element technology: natural coordinates, isoparametric description and numerical integration Together these revolutionized the finite element field in the mid-1960’s §18.1.1 Natural Coordinates Before presenting examples of hexahedron elements, we have to introduce the appropriate natural coordinate system for that geometry The natural coordinates for this geometry are called ξ , η and µ, and are called isoparametric hexahedral coordinates or simply natural coordinates These coordinates are illustrated in Figure 18.1 As can be seen they are very similar to the quadrilateral coordinates ξ and η used in IFEM They vary from -1 on one face to +1 on the opposite face, taking the value zero on the “median” face As in the quadrilateral, this particular choice of limits was made to facilitate the use of the standard Gauss integration formulas §18.1.2 Corner Numbering Rules The eight corners of a hexahedron element are locally numbered 1, The corner numbering rule is similar to that given for the 4-node tetrahedron in Chapter 14 Again the purpose is to guarantee a positive volume (or, more precisely, a positive Jacobian determinant at every point) The transcription of those rules to the hexahedron element is as follows: Chose one starting corner, which is given number 1, and one initial face pertaining to that corner (given a starting corner, there are three possible faces meeting at that corner that may be selected) Number the other corners as 2,3,4 traversing the initial face counterclockwise1 while one looks at the initial face from the opposite one Number the corners of the opposite face directly opposite 1,2,3,4 as 5,6,7,8, respectively “Anticlockwise” in British 18–3 18–4 Chapter 18: HEXAHEDRON ELEMENTS µ η z x y ξ Figure 18.1 The 8-node hexahedron and the natural coordinates ξ , η, µ The definition of the latter is the same for higher order models The definition of ξ , η and µ can be now be made more precise: ξ goes from −1 from (center of) face 1485 to +1 on face 2376 η goes from −1 from (center of) on face 1265 to +1 on face 3487 µ goes from −1 from (center of) on face 1234 to +1 on face 5678 The center of a face is the intersection of the two medians §18.2 THE EIGHT NODE (TRILINEAR) HEXAHEDRON The eight-node hexahedron shown in Figure 18.2 is the simplest member of the hexahedron family It is defined by    1  x   x1     y   y1     z  =  z1     vx   vx1    vy v y1 vz vz1 x2 y2 z2 vx2 v y2 vz2 x3 y3 z3 vx3 v y3 vz3 x4 y4 z4 vx4 v y4 vz4 x5 y5 z5 vx5 v y5 vz5 x6 y6 z6 vx6 v y6 vz6 x7 y7 z7 vx7 v y7 vz7 The hexahedron coordinates of the corners are (see Figure 18.1) 18–4   N (e)  x8   N (e)    y8      z8        vx8     v y8 N8(e) vz8 (18.1) 18–5 §18.3 THE 20-NODE (SERENDIPITY) HEXAHEDRON 19 20 z 16 18 17 13 12 x y 15 11 14 10 Figure 18.2 The 20-node hexahedron element — note node numbering conventions node ξ η µ −1 +1 +1 −1 −1 +1 +1 −1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 The shape functions are N1(e) = 18 (1 − ξ )(1 − η)(1 − µ), N2(e) = 18 (1 + ξ )(1 − η)(1 − µ) N3(e) = 18 (1 + ξ )(1 + η)(1 − µ), N4(e) = 18 (1 − ξ )(1 + η)(1 − µ) N5(e) = 18 (1 − ξ )(1 − η)(1 + µ), N6(e) = 18 (1 + ξ )(1 − η)(1 + µ) N7(e) = 18 (1 + ξ )(1 + η)(1 + µ), N8(e) = 18 (1 − ξ )(1 + η)(1 + µ) (18.2) These eight formulas can be summarized in a single expression: N1(e) = 18 (1 + ξ ξi )(1 + ηηi )(1 + µµi ) where ξi , ηi and µi denote the coordinates of the i th node 18–5 (18.3) 18–6 Chapter 18: HEXAHEDRON ELEMENTS 19 20 z 17 13 x 25 22 21 24 18 12 y 26 16 27 11 15 23 14 10 Figure 18.3 The 27-node hexahedron element — note node numbering conventions §18.3 THE 20-NODE (SERENDIPITY) HEXAHEDRON The 20-node hexahedron is the analog of the 8-node “serendipity” quadrilateral The corner nodes are augmented with 12 side nodes which are usually located at the midpoints of the sides The numbering scheme is illustrated in Figure 18.2 For elasticity applications this element have 20 × = 60 degrees of freedom The 8-node quadrilateral studied in IFEM cannot represent a complete biquadratic expansion in the quadrilateral coordinates ξ and η, that is, the nine terms 1, ξ , η, ξ , , ξ η2 One has to go to the 9-node (biquadratic) quadrilateral to achieve that Likewise, the 20 node hexahedron is incapable of accomodating a full triquadratic expansion in ξ , η and µ; that is 1, ξ , η, µ, η2 , , ξ η2 µ2 A 27-node hexahedron is required for that That element is described in the next section The shape functions of the 20-node hexahedron can be grouped as follows For the corner nodes i = 1, 2, , 8: Ni(e) = 18 (1 + ξ ξi )(1 + ηηi )(1 + µµi )(ξ ξi + ηηi + µµi − 2) (18.4) For the midside nodes i = 9, 11, 17, 19: Ni(e) = 14 (1 − ξ )(1 + ηηi )(1 + µµi ) (18.5) For the midside nodes i = 10, 12, 18, 20: Ni(e) = 14 (1 − η2 )(1 + ξ ξi )(1 + µµi ) (18.6) For the midside nodes i = 13, 14, 15, 16: Ni(e) = 14 (1 − µ2 )(1 + ξ ξi )(1 + ηηi ) 18–6 (18.7) 18–7 §18.5 PARTIAL DERIVATIVES §18.4 THE 27-NODE HEXAHEDRON A 27-node hexahedron can indeed be constructed by adding more nodes: on each face center, and interior node at the hexahedron center See Figure 18.3 In elasticity application such an element has 27 × = 81 degrees of freedom (To be completed) §18.5 PARTIAL DERIVATIVES The calculation of partial derivatives of hexahedron shape functions with respect to Cartesian coordinates follows techniques similar to that discussed for two-dimensional quadrilateral elements in IFEM Only the size of the matrices changes because of the appearance of the third dimension §18.5.1 The Jacobian The derivatives of the shape functions are given by the usual chain rule formulas: ∂ Ni(e) ∂ξ ∂ Ni(e) ∂η ∂ Ni(e) ∂µ ∂ Ni(e) = + + , ∂x ∂ξ ∂ x ∂η ∂ x ∂µ ∂ x ∂ Ni(e) ∂ξ ∂ Ni(e) ∂η ∂ Ni(e) ∂µ ∂ Ni(e) = + + , ∂y ∂ξ ∂ y ∂η ∂ y ∂µ ∂ y (18.8) ∂ Ni(e) ∂ξ ∂ Ni(e) ∂η ∂ Ni(e) ∂µ ∂ Ni(e) = + + ∂z ∂ξ ∂z ∂η ∂z ∂µ ∂z  (e)    ∂ξ ∂η ∂µ  ∂ Ni ∂ Ni(e) ∂ x ∂ x ∂ x  ∂ξ   ∂x  (e)   ∂ N (e)   ∂ξ ∂η ∂µ    ∂ N   = i i   ∂y   ∂y ∂y ∂y    ∂η      ∂ξ ∂η ∂µ ∂ Ni(e) ∂ Ni(e) ∂z ∂z ∂z z The ì matrix that appears in (18.9) is J , the inverse of:   ∂ x ∂ y ∂z  ∂ξ ∂ξ ∂ξ  ∂(x, y, z)  ∂ y ∂z  J= =  ∂x  ∂(ξ, η, µ)  ∂η ∂η ∂η  ∂ x ∂ y ∂z ∂µ ∂µ ∂µ In matrix form  (18.9) (18.10) Matrix J is called the Jacobian matrix of (x, y, z) with respect to (ξ, η, µ) In the finite element literature, matrices J and J−1 are called simply the Jacobian and inverse Jacobian, respectively, although such a short name is sometimes ambiguous The notation J= ∂(x, y, z) , ∂(ξ, η, µ) J−1 = ∂(ξ, η, µ) ∂(x, y, z) (18.11) is standard in multivariable calculus and suggests that the Jacobian may be viewed as a generalization of the ordinary derivative, to which it reduces for a scalar function x = x(ξ ) 18–7 18–8 Chapter 18: HEXAHEDRON ELEMENTS §18.5.2 Computing the Jacobian Matrix The isoparametric definition of hexahedron element geometry is x = xi Ni(e) , y = yi Ni(e) , z = z i Ni(e) , (18.12) where the summation convention is understood to apply over i = 1, 2, n, in which n denotes the number of element nodes Differentiating these relations with respect to the hexahedron coordinates we construct the matrix J as follows:  ∂ Ni(e) ∂ Ni(e) ∂ Ni(e) yi ∂ξ z i ∂ξ   xi ∂ξ  (e) (e)    ∂ Ni(e) ∂ N ∂ N i i (18.13) J =  xi  y z i i  ∂η ∂η ∂η    ∂ N (e) ∂ N (e) ∂ N (e) xi ∂µi yi ∂µi z i ∂µi Given a point of hexahedron coordinates (ξ, η, µ) the Jacobian J can be easily formed using the above formula, and numerically inverted to form J−1  REMARK 18.1 The inversion formula for a matrix of order is A= a11 a21 a31 a12 a22 a32 a13 a23 a33 , A −1 = |A| A11 A21 A31 A12 A22 A32 A13 A23 A33 , (18.14) where A11 = a22 a33 − a23 a32 , A22 = a33 a11 − a31 a13 , A33 = a11 a22 − a12 a21 , A12 = a23 a31 − a21 a33 , A23 = a31 a12 − a32 a11 , A31 = a12 a23 − a13 a22 , A21 = a32 a13 − a12 a33 , A32 = a13 a21 − a23 a11 , A13 = a21 a22 − a31 a22 , |A| = a11 A11 + a12 A21 + a13 A31 (18.15) (The determinant can in fact be computed in different ways.) §18.6 THE STRAIN DISPLACEMENT MATRIX Having obtained the shape function derivatives, the matrix B for a hexahedron element displays the usual structure for 3D elements:  ∂/∂ x   B = DΦ =   ∂/∂ y 0 ∂/∂ y ∂/∂ x ∂/∂z  0  q  ∂/∂z  q  0 ∂/∂ 18–8  0 q qx    =  qy  qz qy qx qz  0   qz     qy qx (18.16) 18–9 §18.7 STIFFNESS MATRIX EVALUATION where q = [ N1(e) ··· Nn(e) ] ∂ N1(e) ∂ Nn(e) · · · ∂ x(e) ∂x ∂ N ∂ Nn(e) qy = · · · ∂y ∂y (e) ∂ Nn(e) qz = ∂ N1 · · · ∂z ∂z are row vectors of length n, n being the number of nodes in the element qx = §18.7 STIFFNESS MATRIX EVALUATION The element stiffness matrix is given by K(e) = V (e) BT EB d V (e) (18.17) As in the two-dimensional case, this is replaced by a numerical integration formula which now involves a triple loop over conventional Gauss quadrature rules Assuming that the stress-strain matrix E is constant over the element, K(e) = p1 p2 p3 wi w j wk BiTjk EBi jk Jik (18.18) i=1 j=1 k=1 Here p1 , p2 and p3 are the number of Gauss points in the ξ , η and µ direction, respectively, while Bi jk and Ji j are abbreviations for Bi jk ≡ B(ξi , η j , µk ), Jik ≡ detJ(ξi , η j , µk ) (18.19) Usually the number of integration points is taken the same in all directions: p = p1 = p2 = p3 The total number of Gauss points is thus p Each point adds at most to the stiffness matrix rank The minimum rank-sufficient rules for the 8-node and 20-node hexahedra are p = and p = 3, respectively REMARK 18.2 The computation of consistent node forces corresponding to body forces is straightforward The treatment of prescribed surface tractions such as pressure, presents, however, some computational difficulties because hexahedron faces are not generally plane §18.7.1 Selecting the Integration Rule Usually the number of integration points is taken the same in all directions: p = p1 = p2 = p3 The total number of Gauss points is thus p Each point adds at most to the stiffness matrix rank For the 8-node hexahedron this rule gives p = because 23 × = 48 > 24 − For other configurations see Exercise 18.3 18–9 18–10 Chapter 18: HEXAHEDRON ELEMENTS Homework Exercises for Chapter 18 Hexahedron Elements EXERCISE 18.1 [A:20] Find the shape functions associated with the 16-node hexahedron depicted in Figure E18.1(a) for node points and (This kind of element is historically important as a pit stop on the way the “degenerated solid” thick-shell elements developed in the late 1960s.) (a) 16 (b) 15 12 18 14 13 12 14 13 11 15 16 11 17 10 10 Figure E18.1 (a): 16-node hexahedron for Exercise 18.1; (b): 18-node hexahedron for Exercise 18.2 EXERCISE 18.2 [A:20] Find the shape functions associated with the 18-node hexahedron depicted in Figure E18.1(b) for node points 1, and 17 EXERCISE 18.3 [A:15] Which minimum integration rules of Gauss-product type gives a rank sufficient stiffness matrix for (a) the 20-node hexahedron, (b) the 27-node hexahedron, (c) the the 16-node hexahedron of Exercise 18.1 and (d) the 18-node hexahedron of Exercise 18.2 For the last two, would a formula containing less Gauss sample points in the µ direction (for example: × × work, at least on paper? 18–10 ... IFEM Only the size of the matrices changes because of the appearance of the third dimension §18.5.1 The Jacobian The derivatives of the shape functions are given by the usual chain rule formulas:... hexahedron coordinates (ξ, η, µ) the Jacobian J can be easily formed using the above formula, and numerically inverted to form J−1  REMARK 18.1 The inversion formula for a matrix of order is A=... matrix for (a) the 20-node hexahedron, (b) the 27-node hexahedron, (c) the the 16-node hexahedron of Exercise 18.1 and (d) the 18-node hexahedron of Exercise 18.2 For the last two, would a formula