The proper generalized decomposition for advanced numerical simulations ch25 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.
25 Kirchhoff Plates: BCs and Variational Forms 25–1 25–2 Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS TABLE OF CONTENTS Page §25.1 INTRODUCTION 25–3 §25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE §25.2.1 Conjugate Quantities §25.2.2 The Modified Shear §25.2.3 Corner Forces §25.2.4 Common Boundary Conditions §25.2.5 Strong Form Diagram 25–3 25–3 25–4 25–5 25–6 25–7 §25.3 THE TOTAL POTENTIAL ENERGY PRINCIPLE §25.3.1 The TPE Functional §25.3.2 Finite Element Conditions 25–7 25–7 25–8 §25.4 THE HELLINGER-REISSNER PRINCIPLE §25.4.1 Finite Element Conditions 25–8 25–9 §25.5 THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE 25–10 §25.5.1 The dV Functional 25–10 §25.5.2 Finite Element Conditions 25–10 25–2 25–3 §25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE §25.1 INTRODUCTION In this Chapter we continue the discussion of the governing equations of Kirchhoff plates with the consideration of boundary conditions (BCs) When plates and shells are numerically idealized by finite elements the proper modeling of boundary conditions can be a difficult subject Two factors contribute to this First, displacement derivatives in the form of rotations are now involved in the kinematic boundary conditions Second, the correlation between physical support conditions and mathematical B.C can be tenous Some mathematical BC used in practice are nearly impossible to reproduce in the laboratory, let alone on an actual structure §25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE One of the mathematical difficulties associated with this plate model is the “Poisson paradox” resolved by Kirchhoff: • The plate deflection satisfies a fourth order partial differential equation (PDE), which is the biharmonic equation ∇ w = q/D for an isotropic homogeneous plate • A fourth order PDE can only have two boundary conditions at each boundary point • But three conjugate quantities: normal moment, twist moment and transverse shear appear naturally there The reduction from three to two requires variational methods But it is not necessary to look at a complete functional The procedure can be explained directly through the external boundary work §25.2.1 Conjugate Quantities Conside a Kirchhoff plate of general shape as in Figure 25.1(a) Assume that the boundary is smooth, that is, contains no corners Under those assumptions the exterior normal n and tangential direction s at each boundary point B are unique, and form a system of local Cartesian axes The kinematic quantities referred to these local axes are w, ∂w = −θs , ∂n ∂w = θn , ∂s (25.1) where θn and θs denote the rotations of the midsurface at B about axes n and t, respectively, see Figure 25.1(b) The work conjugate static quantities, shown in Figure 25.1(c), are Qn , Mnn , Mns , (25.2) respectively By conjugate it is meant that the boundary work can be expressed as the line integral WB = Q n w + Mns ∂w ∂w + Mnn ds = ∂s ∂n Q n w + Mns θn − Mnn θs ds (25.3) where ds ≡ d denotes the differential boundary arclength This integral appears naturally in the process of forming the energy functionals of the plate Given the configuration of W B , it appears at 25–3 25–4 Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS dx (b) θs y (a) s dy ds B n θn Γ x A Ω B Qx (c) y Mxy Qy Mxx Qn Myx x s B Myy Mnn Mns n Figure 25.1 BCs at a smooth boundary point B of a Kirchhoff plate: n = external normal, s = tangential direction (a) boundary traversed in the counterclockwise sense (looking down from +z) leaving the plate proper on the left; (b) shows kinematic quantities θs and θn ; (c) shows force-moment quantities Q n , Mnn and Mns on the boundary face first sight as if three boundary conditions can be assigned at each boundary point, taken from the conjugate sets (25.1) and (25.2) For example: Simply supported edge Free edge w=0 Qn = θn = Mns = Mnn = 0, Mnn = (25.4) The boundary conditions for a free edge were indeed expressed by Poisson in this form.1 As noted above, this is inconsistent with the order of the PDE Kirchhoff showed2 that three conditions are too many and in fact only two are independent §25.2.2 The Modified Shear The reduction to two independent conjugate pairs may be demonstrated through integration of (25.3) by parts with respect to s, along a segment AB of the boundary : W B | BA = B Qn − A ∂w ∂ Mns w + Mnn dt + Mns w| BA ∂s ∂n Introducing the modified shear3 Vn = Q n − ∂ Mns , ∂s See e.g., I Todhunter and K Pierson, History of Theory of Elasticity, Vol I G Kirchhoff, publications cited in §24.2 Also called Kirchhoff equivalent force, or Kirchhoffische Ersatzkrăafte 254 (25.5) (25.6) 255 Đ25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE s s Rc _ Mns + Mns C Figure 25.2 Effect of of modified shear (Kirchhoff shear) at a plate corner: the force-pairs not cancel, producing a corner force Rc we may rewrite (25.5) as W B | BA = B Vn w + Mnn A ∂w dt + Mns w| BA ∂n (25.7) This transformation reduces the conjugate quantities to two work pairs: Vn , w Mnn , and ∂w = −θs ∂n (25.8) §25.2.3 Corner Forces The last term in (25.7) deserves analysis First consider a plate with smooth boundary as in Figure 25.1 Assuming that Mns is continuous over , and we go completely “around” the plate so that A ≡ B, (25.9) Mns w| BA = Next consider the case of a plate with a corner C as in Figure 25.2 At C the twisting moment jumps − + from, say, Mns to Mns The transverse displacement w must be continuous (if fact, C continuous) Place A and B to each side of C, so that A → C from the minus side while C ← B from the plus side Then + − − Mns )w, Mns | BA = Rc w = (Mns with + − Rc = Mns − Mns (25.10) This jump in the twisting moment is called the corner force Rc ; see Figure 25.3 Note that Rc has the physical dimension of force, because the twisting moment is a moment (force times length) per unit length Simple Support, No Edge Anchorage Figure 25.3 Manifestation of modified shear as corner lifting forces 25–5 25–6 Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS Free edge ;; ;; ;;;; ;;;; ; water Clamped (fixed) edges "Point support" for reinforced concrete slab Figure 25.4 Boundary condition examples REMARK 25.1 The physical interpretation of modified shears and of corner forces is well covered in Timoshenko and Woinowsky-Krieger.4 Suffices to say that if a plate corner is constrained not to move laterally, a concentrated force Rc called the corner reaction, appears If the corner is not held down the reaction cannot physically manifest and the plate will have a tendency to move away from the support This is the source of the well known “corner lifting” phenomenon that may be observed on a laterally loaded square plate with simply supported edges that not prevent lifting See Figure 25.3 This effect does not appear if the edges meeting at C are free or clamped, because if so the twist moment Mns on both sides of the corner point are zero §25.2.4 Common Boundary Conditions Below we state homogeneous boundary conditions frequently encountered in Kirchhoff plates as selected combinations of the conjugate quantities (25.8) Some BCs are illustrated in the structures depicted in Figure 25.4 Clamped or Fixed Edge (with s along edge): ∂w = −θs = ∂n (25.11) w = 0, Mnn = (25.12) Vn = 0, Mnn = (25.13) w = 0, Simply Supported Edge (with s along edge): Free Edge (with s along edge): Theory of Plates and Shells monograph cited in previous Chapter 25–6 25–7 §25.3 ^ w=w θs = ^θs Prescribed deflections & rotations ^s ^ θ w, Displacement BCs THE TOTAL POTENTIAL ENERGY PRINCIPLE Lateral load q Deflection w Kinematic κ=Pw in Ω Γ PT M = q in Ω Equilibrium Constitutive M=Dκ Curvatures κ in Ω Ω Bending moments M ^ Mnn= M nn Vn = V^n Force BCs Prescribed moments & shears ^ , V^ M nn n Figure 25.5 The Strong Form diagram for the Kirchhoff plate, including boundary conditions Symmetry Line (with s along line): Vn = 0, ∂w = −θs = ∂n (25.14) Point Support: w=0 (25.15) Non-homogeneous boundary conditions of force type involving prescribed normal moment Mˆ nn or prescribed modified shear Vˆn , are also quite common in practice Non-homogeneous B.C involving prescribed nonzero transverse displacements or rotations are less common §25.2.5 Strong Form Diagram The Strong Form diagram of the governing equations, including boundary conditions, for the Kirchhoff plate model is shown in Figure 25.5 We are now ready to present several energy functionals of the Kirchhoff plate that have been used in the construction of finite elements §25.3 THE TOTAL POTENTIAL ENERGY PRINCIPLE The only master field is the transverse displacement w The departure Weak Form is shown in Figure 25.5 The weak links are the internal equilibrium equations and the force boundary conditions §25.3.1 The TPE Functional Proceeding as explained in previous chapters one arrives at the functional The TPE functional with the conventional forcing potential is TPE [w] = UTPE [w] − WTPE [w] (25.16) The internal energy is UTPE [w] = (Mw )T κw = (κw )T D κw d 25–7 = (wPT )D (Pw) d , (25.17) 25–8 Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS Prescribed deflections & rotations ^s ^ θ w, ^ w=w θs = ^θs Lateral load q Deflection w Master w Displacement Rotations θ BCs Kinematic κw = P w in Ω Ω Equilibrium Constitutive Slave Curvatures κw Γ Mw= D κw Bending moments Mw in Ω Slave Force BCs Prescribed moments & shears ^ , V^ M nn n Figure 25.6 The Weak Form departure point to derive the TPE variational principle for a Kirchhoff plate where P = [ ∂ /∂ x ∂ /∂ y 2 ∂ /∂ x∂ y ]T is the curvature-displacement operator The groupings Pw in the last of (25.17) emphasize that P is to be applied to w to form the slave curvatures κw The external work WTPE [w] is more complicated than in plane stress and solids It is best presented as the sum of three components These are due to apply lateral loads, applied edge moments and transverse shears, and to corner loads, respectively: WTPE [w] = Wq [w] + W B [w] + WC [w], (25.18) The first two terms apply to all plate geometries and are Wq [w] = qwd , (Vˆn w − Mˆ nn θsw ) d W B [w] = (25.19) VM The last term: WC arises if the plate has j = 1, 2, , n c corners at which the displacement w j is not prescribed If so, nc WC = nc Rcj w j = j=1 + − ( Mˆ ns − Mˆ ns ) w (25.20) j=1 §25.3.2 Finite Element Conditions The variational index of the TPE functional is m = 2, since second derivatives of w (the curvatures) appear in UTPE The convergence conditions for finite elements derived using this principle are: Completeness The assumed w over each element should reproduce exactly all {x, y} polynomials of order ≤ Continuity The assumed w should be C continuous over the FEM mesh The second condition is not easy to satisfy using standard polynomial assumptions These difficulties have motivated the development of various techniques to alleviate the continuity requirement 25–8 25–9 §25.4 Prescribed deflections & rotations ^s ^ θ w, THE HELLINGER-REISSNER PRINCIPLE ^ w=w θs = ^θs Lateral load q Deflection w Master w Displacement Rotations θ BCs Kinematic Slave κ =Pw in Ω w Γ Equilibrium Curvatures κw Constitutive Slave Ω Curvatures κM Mw= D κw Bending moments M in Ω Master Force BCs Prescribed moments & shears ^ , V^ M nn n Figure 25.7 The Weak Form departure point to derive the HR variational principle for a Kirchhoff plate §25.4 THE HELLINGER-REISSNER PRINCIPLE The Weak Form useful as departure point for the HR principle is shown in Figure 25.7 Both the transverse displacement w and the bending moment field M are chosen as master fields The weak links are the internal equilibrium equations, The HR functional with the conventional forcing potential is HR [w, M] = UHR [w, M] − WHR [w, M] (25.21) The internal energy is UHR [w, M] = (MT κw − 12 MT D−1 M) d = (MT κw − U ∗ ) d (25.22) Here U ∗ = 12 MT D−1 M) is the complementary energy density (per unit of plate area) written in terms of the bending moments Integration of this over gives the total complementary energy U ∗ The external work WHR is the same as for the TPE principle treated in the previous section §25.4.1 Finite Element Conditions The variational indices of the HR functional are m w = for the transverse deflection and m M = for the bending moments Consequently the completeness and continuity conditions for w are the same as for the TPE, and nothing is gained by going to the more complicated functional 25–9 25–10 Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS Prescribed deflections & rotations ^s ^ θ w, ^ w=w θs = ^θs Lateral load q Deflection w Master w Displacement Rotations θ BCs Kinematic Γ κ =Pw in Ω w Constitutive Slave M =Dκ in Ω w Curvatures κw w Constitutive Mκ = D κ Master Curvatures κ in Ω Slave Bending moments Mw Bending moments Mκ Ω Equilibrium Slave Force BCs Prescribed moments & shears ^ , V^ M nn n Figure 25.8 The Weak Form departure point to derive the curvature-displacement de Veubeke variational principle for a Kirchhoff plate It is possible to balance the variational indices so that m w = m M = by integrating the previous form by parts once The resulting principle was exploited by Herrmann5 to construct a plate element with linearly varying w, Mx x , M yy , Mx y This element, however, was disappointing in accuracy Furthermore enforcing moment continuity can be physically wrong Progress in the construction of elements of this type was achieved later using hybrid principles §25.5 THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE This kind of principle (for elastic solids) was introduced by de Veubeke, and functionals will be accordingly identified by a dV subscript §25.5.1 The dV Functional In this case both the transverse displacement w and the curvatures field κ are chosen as master fields The departure Weak Form is shown in Figure 25.8 dV [w, κ] = UdV [w, κ] − WdV [w, κ] (25.23) The internal functional is dV [w, κ] = (κT Mw − 12 κT Dκ) d (25.24) The external work is the same as for TPE L R Herrmann, A bending analysis for plates, in Proceedings 1st Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, Air Force Institute of Technology, Dayton, Ohio, pp 577-604, 1966 25–10 25–11 §25.5 THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE §25.5.2 Finite Element Conditions The variational indices of the dV functional is m w = for the transverse displacement and m κ = for the curvatures Consequently the completeness and continuity conditions for w are the same as for the TPE, as nothing is gained by going to the more complicated functional To get a practical scheme that reduces the continuity order it is necessary to proceed to hybrid principles 25–11 ... as for the TPE principle treated in the previous section §25.4.1 Finite Element Conditions The variational indices of the HR functional are m w = for the transverse deflection and m M = for the. .. PRINCIPLE The Weak Form useful as departure point for the HR principle is shown in Figure 25.7 Both the transverse displacement w and the bending moment field M are chosen as master fields The weak... concentrated force Rc called the corner reaction, appears If the corner is not held down the reaction cannot physically manifest and the plate will have a tendency to move away from the support This is the