The proper generalized decomposition for advanced numerical simulations ch02 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.
2 Decomposition of Poisson Problems 2–1 2–2 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS TABLE OF CONTENTS Page §2.1 INTRODUCTION 2–3 §2.2 THE POISSON EQUATION §2.3 STEADY-STATE LINEAR HEAT CONDUCTION §2.3.1 The Field Equations §2.3.2 The Boundary Conditions §2.3.3 Summary of Governing Equations §2.3.4 Tonti Diagrams §2.3.5 Alternative Notations 2–3 2–4 2–5 2–6 2–6 2–7 2–7 §2.4 STEADY POTENTIAL FLOW §2.5 ELECTROSTATICS 2–9 2–10 §2.6 *MAGNETOSTATICS 2–11 EXERCISES 2–13 2–2 2–3 §2.2 THE POISSON EQUATION §2.1 INTRODUCTION In the first Chapter it was emphasized that the classical formulation of mathematical models in mechanics and physics leads to the Strong Form (SF) These field equations are ordinary or partial differential equations in space or spacetime in a primary variable They are complemented by boundary and/or initial conditions Field equations and boundary plus initial conditions are collectively called the governing equations The distinguishing property of the SF is that governing equations and conditions hold at each point of the problem domain Passing from the Strong Form to Weak and Variational Forms is simplified if the governing equations are presented through a scheme called Tonti decompositions.1 Such schemes introduce two auxiliary variables that often have physical significance One is called the intermediate variable and the other the flux variable Examples of such variables are stresses, strains, pressures and heat fluxes The equations that connect the primary and auxiliary variables in the decomposed SF are called Strong Links or Strong Connectors Tonti decompositions offer two important advantages for further development: (I) The construction of various types of Weak and Variational Forms can be graphically explained as weakening selected links (II) The interpretation of the so-called natural boundary conditions is facilitated This Chapter illustrates the construction of the Tonti decomposition for boundary value problems modeled by the scalar Poisson’s equation We start with these problems because the governing equations are considerable simpler than for the elasticity problem of structural and solid mechanics The simplicity is due to the fact that the primary variable is a scalar function whereas the intermediate variables are vectors On the other hand, in elasticity the primary variable: displacements, is a vector, whereas the intermediate variables: strains and stresses, are tensors Despite this simplicity the Poisson equation governs several interesting problems in engineering and physics The next Chapter illustrates the construction of Weak and Variational Forms §2.2 THE POISSON EQUATION Many steady-state application problems in mechanics and electromagnetics can be modeled by the generalized Poisson’s partial differential equation This includes the famous Laplace equation as a special case Suppose that u = u(x1 , x2 , x3 ) is the primary scalar function that solves a linear, steady-state (time-independent) application problem involving an isotropic medium The problem is posed in a three-dimensional space spanned by the Cartesian coordinates x1 , x2 , x3 The generalized Poisson’s equation is ∇ · (ρ∇u) = s, (2.1) where the first ∇ is the divergence operator, the second ∇ is the gradient operator, s is a given source function, and ρ is a constitutive coefficient (This coefficient becomes a tensor ρi j in anisotropic media) A name suggested by a graphical representation introduced by the mathematician E Tonti 2–3 2–4 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS Both ρ and s may depend on the spatial coordinates, that is, ρ = ρ(x1 , x2 , x3 ) and s = s(x1 , x2 , x3 ) Equation (2.1) must be complemented by appropriate boundary conditions These are examined in further detail in connection with the specific examples in §2.3 If ρ is constant in space, (2.1) reduces to the standard Poisson’s equation ρ∇ u = s (2.2) where ∇ is the Laplace operator Furthermore if the source term s vanishes this reduces to the familiar Laplace’s equation ∇ u = (2.3) Solutions of (2.3) are called harmonic functions, which have been extensively studied over the past two centuries The extension of the foregoing equations to an unknown vector function u is straightforward In such a case the first ∇ in (2.1) is the gradient operator and the second ∇ the divergence operator REMARK 2.1 In unabridged component notation the Poisson’s equation (2.1) in one, two, and three dimensions takes the following form: ∂ ∂u ρ = s, ∂ x1 ∂ x1 ∂ ∂ x1 ρ ∂u ∂ x1 + ∂ ∂ x1 ρ ∂u ∂ x1 + ∂ ∂ x2 ρ ∂u ∂ x2 = s, ∂ ∂ x2 ρ ∂u ∂ x2 + ∂ ∂ x3 ρ ∂u ∂ x3 = s (2.4) If k is not space dependent: ρ ∂ 2u = s, ∂ x12 ρ ∂ 2u ∂ 2u + ∂ x12 ∂ x22 = s, ρ ∂ 2u ∂ 2u ∂ 2u + + ∂ x12 ∂ x22 ∂ x32 = s (2.5) By specializing the primary variable u to various physical quantities, we obtain models for various problems in mechanics, thermomechanics and electromagnetics Three specific problems: thermal conduction, potential flow, electro and magnetostatics are examined below Other applications are given as Exercises §2.3 STEADY-STATE LINEAR HEAT CONDUCTION Consider a thermally conducting isotropic body of volume V that obeys Fourier’s law of heat conduction, as illustrated in Figure 2.1 The body is bounded by a surface S with external unit normal n The body is in thermal equilibrium, meaning that the temperature distribution T = T (x1 , x2 , x3 ) is independent of time The temperature is the primal variable of this formulation If the body is thermally isotropic, the ρ of the previous section becomes the thermal conductivity coefficient k, with a − sign to account for the positive flux sense definition wThis coefficient may be a function of position 2–4 2–5 §2.3 STEADY-STATE LINEAR HEAT CONDUCTION n Sq : q^n = q x3 x1 x2 Volume V ^ Heat source production in V : h specified per unit of volume ST : T = T Figure 2.1 A heat conducting body obeying Fourier’s conduction, in thermal equilibrium The source field called s in the previous section is the distributed heat production h = h(x1 , x2 , x3 ) in V measured per unit of volume This heat may be generated, for instance, by combustion or by a chemical reaction.2 A negative h would indicate a volumetric heat dissipation or “sink.” §2.3.1 The Field Equations The temperature gradient vector is called g = ∇T , which written in full is g1 g2 g3 = ∂ T /∂ x1 ∂ T /∂ x2 ∂ T /∂ x3 (2.6) The temperature gradient along a direction d defined by the unit vector d is ∂ T /∂d = g · d = gT d In particular the boundary-normal temperature gradient is ∂ T /∂n = g · n = gT n evaluated on the surface S The heat flux vector q is defined by the constitutive equation q = −kg = −k∇T , which is Fourier’s law of heat conduction In full this is q1 q2 q3 = −k g1 g2 g3 (2.7) The heat flux along a direction d defined by the unit vector d is denoted by qd = q · d = qT d This is a scalar that characterizes the transport of thermal energy along that direction, and is measured in heat units per unit area In particular, the boundary-normal heat flux is qn = q · n = qT n evaluated on S In the human body, the heat source are calories produced from food intake 2–5 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS 2–6 h T div q + h = in V g = grad T in V g q = − k g in V q Figure 2.2 Tonti diagram for steady-state heat conduction problem, showing only the field equations The balance equation, which characterizes steady-state thermal equilibrium, is div q + h = Written in full: ∂q2 ∂q2 ∂q1 + + + h = (2.8) ∂ x1 ∂ x2 ∂ x3 Equations (2.6), (2.7) and (2.8) complete the field equations of the heat conduction problem §2.3.2 The Boundary Conditions The classical boundary conditions for this problem are of two types: The temperature T is prescribed to be equal to Tˆ over a portion ST of the boundary S (ST is colored red in Figure 2.1) The boundary-normal heat flux qn = q.n = −k(∂ T /∂n) is prescribed to be equal to qˆn over the complementary portion Sq of the boundary S : ST ∪ Sq (Sq is colored blue in Figure 2.1) Other boundary conditions that occur in practice are those due to radiation and to convection Those are more complex (in fact, thay are nonlinear) and are not considered here §2.3.3 Summary of Governing Equations The field equations, expressed in direct notation, are now summarized and labeled: KE: CE: BE: ∇T = g − kg = q ∇ ·q+h =0 in V, in V, in V (2.9) The kinematic equation (KE) is simply the definition of the temperature gradient vector g The constitutive equation (CE) is Fourier’s law of thermal conduction The balance equation (BE) is the law of thermal equilibrium: the heat flux gradient must equal to the heat created (or dissipated) per unit volume These three labels: KE, CE and BE, will be used throughout this course for wide classes of problems governed by differential equations in space variables Fields g and q are called the intermediate variable and the flux variable, respectively 2–6 2–7 §2.3 T^ ^ T=T on S T STEADY-STATE LINEAR HEAT CONDUCTION h T g = grad T in V div q + h = in V g q = − k g in V q q n = q.n = q^ q^ on Sq Strong connection Data field Unknown field Figure 2.3 Expanded Tonti diagram for the steady-state heat conduction problem, showing BCs Elimination of the intermediate variables g and q in (2.9) yields the scalar Poisson’s equation ∇ · (ρ∇T ) = h (2.10) This shows that steady-state Fourier heat conduction pertains to the “Poisson-problem” class typified by (2.1), in which ρ → −k, u → T and s → h The two classical boundary conditions are labeled as PBC: T = Tˆ on ST , FBC: q n = qn = qˆn on Sq T (2.11) Here labels PBC and FBC denote primary boundary conditions and flux boundary conditions, respectively These labels will be also used throughout the course The set of field equations: KE, CE, BE, and boundary conditions: PBC and FBC, are collectively called the governing equations These equations constitute the statement of the mathematical model for this particular problem This formulation is called a boundary value problem, or BVP §2.3.4 Tonti Diagrams A convenient graphical representation of the three field equations is the so-called Tonti-diagram, which is drawn in Figure 2.2 This diagram can be expanded as illustrated in Figure 2.3 to include the boundary conditions Graphical conventions for this expanded diagram are explained in this figure The term “strong connection” for a relation means that it applies point by point A “data field” is one that is given as part of the problem specification The expanded Tonti diagram has been found to be more convenient from the instructional standpoint than the reduced diagram, and will be adopted from now on Figure 2.4 shows the generic names of the components of the expanded Tonti diagram 2–7 2–8 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS Specified primary variable Primary boundary conditions Primary variable FIELD EQUATIONS Balance or equilibrium equations Kinematic equations Intermediate variable Source function Constitutive equations Flux variable Flux boundary conditions Specified flux variable Figure 2.4 Names of the components of the expanded Tonti diagram §2.3.5 Alternative Notations To facilitate comparison with reference works, the governing equations are restated below in three alternative forms: in ‘grad/div’ notation, in compact indicial notation and in full (unabridged) component notation For the second form the summation convention is implied Indices run from through for the three-dimensional case The range is reduced to or if the number of space dimensions is reduced to two and one, respectively Matrix/vector form: KE: CE: grad T = g − kg = q in V, in V, BE: div q + h = T = Tˆ in V, PBC: FBC: q.n = qn = qˆn , (2.12) on ST , on Sq Indicial form: KE: CE: BE: PBC: FBC: T,i = gi − kgi = qi qi,i + h = T = Tˆ qi n i = qn = qˆn , 2–8 in V, in V, in V, on ST , on Sq (2.13) 2–9 §2.4 STEADY POTENTIAL FLOW Unabridged: KE: CE: BE: PBC: FBC: ∂T ∂T ∂T = g1 , = g2 , = g3 , ∂ x1 ∂ x2 ∂ x3 − kg1 = q1 , −kg2 = q2 , −kg3 = q3 , ∂q2 ∂q3 ∂q1 + + +h =0 ∂ x1 ∂ x2 ∂ x3 T = Tˆ q1 n + q2 n + q3 n = qˆn in V, in V, in V, (2.14) on ST , on Sq §2.4 STEADY POTENTIAL FLOW As next example consider the potential flow of a fluid of mass density ρ that occupies a volume V The fluid volume is bounded by a surface S with external unit normal n The flow is characterized by the velocity field 3-vector v(x1 , x2 , x3 ), which is independent of time For irrotational flow this field can be expressed as the gradient v = −∇φ of a scalar function φ(x1 , x2 , x3 ) called the velocity potential This potential is chosen as primal variable Note the physical contrast with the thermal conduction problem discussed in §2.3 In heat conduction the primal field — the temperature — has immediate physical meaning whereas the temperature gradient g is a convenient intermediate variable On the other hand, in potential flow the field of primary significance — fluid velocity — is an intermediate variable whereas the primal field — the velocity potential — has no physical significance Despite this contrast the two problems share the same mathematical formulation as explained below The forcing and boundary conditions are as follows: The source field is σ , the fluid mass production per unit of volume Such production is rare in applications Thus for most potential flow problems σ = The potential φ is prescribed to be equal to φˆ over a portion Sφ of the boundary S The fluid momentum density m n = ρv.n is prescribed to be equal to mˆn over the complementary portion Sm of the boundary S : Sφ ) ∩ Sm In practice the most common boundary condition is that of prescribed normal velocity v.n = = vˆn This can be easily transformed to the prescribed momentum density B.C on multiplying by the density Mathematically the momentum density B.C is the correct one The field equations, expressed in direct notation, are: KE: CE: BE: − ∇φ = v ρv = m ∇ ·m=σ in V, in V, in V In fluid mechanics, potential flow is short for steady barotropic irrotational flow of a perfect fluid 2–9 (2.15) Chapter 2: DECOMPOSITION OF POISSON PROBLEMS 2–10 The kinematic equation (KE) is simply the definition of the velocity potential The constitutive equation (CE) is the definition of momentum density The balance equation (BE) expresses conservation of mass Elimination of the intermediate variables v and m in (2.15) yields the scalar Poisson’s equation ∇ · (ρ∇φ) = σ (2.16) This shows that steady potential flow pertains to the “Poisson-problem” class (2.1), in which u → φ and s → σ As noted above, usually σ = whereas ρ is constant, whereupon (2.16) reduces to the Laplace’s equation ∇ φ = The boundary conditions are: PBC: FBC: φ = φˆ m.n = m n = mˆ n , on Sφ , on Sm (2.17) It should be obvious now that steady potential flow and steady heat conduction are mathematically equivalent problems, despite the great disparity in the physical interpretation of primal and intermediate quantities §2.5 ELECTROSTATICS Electrostatics is concerned with the calculation of the steady-state electrical field 3-vector E(x1 , x2 , x3 ) in a volume V filled by a dielectric material or medium of permittivity (this property measures the inductive capacity of the medium; it is also called the dielectric constant) As in the case of potential flow, E is not the primal field but is derived from the electric potential (x1 , x2 , x3 ) as E = −∇ Thus E plays the role of intermediate variable The flux-like variable is the 3-vector D = E, which receives the names of electric field intensity or the electric flux density The forcing and boundary conditions are as follows: The source field is ρ, the electric charge per unit of volume (This symbol should not be confused with mechanical density) For many electrostatic problems all charges migrate to the surface S, thus ρ = in the volume The potential The normal electric flux Dn = D.n is prescribed to be equal to Dˆ n over the complementary portion S D of the boundary S : S ) ∩ S D is prescribed to be equal to ˆ over a portion S of the boundary S The electric potential has more physical significance than the (mathematically equivalent) velocity potential in potential flow In electric circuits this potential can be directly measured as voltage Similarly the flux condition has direct physical interpretation as electric flow, or current Thus both boundary conditions are physically important The field equations, expressed in direct notation, are: 2–10 2–11 §2.6 *MAGNETOSTATICS KE: −∇ =E CE: BE: E=D ∇ ·D=ρ in V, in V, in V (2.18) The kinematic equation (KE) is the definition of the electric potential The constitutive equation (CE) relates electric intensity and flux through the dielectric constant The balance equation (BE) expresses conservation of charge (this last relation is also called Gauss’ law and is one of the famous Maxwell equations) Elimination of the intermediate variables E and D in (2.18) yields the scalar Poisson’s equation ∇ · ( ∇ ) = −ρ (2.19) This shows that electrostatics pertains to the “Poisson-problem” class Often ρ = and constant, whereupon (2.19) reduces to the Laplace’s equation ∇ = is The boundary conditions are: = ˆ PBC: D.n = Dn = Dˆ n , FBC: on S , on S D (2.20) §2.6 *MAGNETOSTATICS Magnetostatics is concerned with the calculation of the steady-state magnetic flux density 3-vector B(x1 , x2 , x3 ) in a volume V filled by a material or medium of permeability µ The magnetic field B is a solenoidal vector (meaning that its divergence is zero) Thus it can be derived from the 3-vector magnetic potential A(x1 , x2 , x3 ) as B = ∇ × A Hence B plays the role of intermediate variable, but unlike the three previous examples, the primal variable A is a vector and not a scalar The flux-like variable is the 3-vector H = (1/µ)B, which receives the name of magnetic field intensity The forcing and boundary conditions are as follows: The source field is J, the electric current density, which is a 3-vector The quantity A × n = A×n is prescribed to be equal to Aˆ ×n over a portion S A of the boundary S The quantity H×n = H × n is prescribed to be equal to Hˆ ×n over the complementary portion S H of the boundary S : S A ) ∩ S H The field equations, expressed in direct notation, are: KE: ìA=B in V, CE: à1 B = H in V, BE: ∇ ×H=J in V 2–11 (2.21) 2–12 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS The kinematic equation (KE) is the definition of the magnetic potential The constitutive equation (CE) relates magnetic intensity and flux through the permeability constant The balance equation (BE) expresses conservation of current (this last relation is one of the famous Maxwell equations) Elimination of the intermediate variables B and H in (2.21) yields ∇ × (µ−1 ∇ × A) = J, (2.22) This can be transformed into a vector Poisson’s equation given in any book on field electromagnetics Finally, the boundary conditions are: PBC: A × n = A×n = Aˆ ×n on S A , FBC: H × n = H×n = Hˆ ×n , on S H 2–12 (2.23) 2–13 Exercises Homework Exercises for Chapter Decomposition of Poisson Problems EXERCISE 2.1 [A:5] Show that div is the transpose of grad when these operators are treated as vectors EXERCISE 2.2 [A:20=15+5] A bar of length L, elastic modulus E and variable cross-sectional area A(x) is aligned along the x axis, extending from x = through x = L The bar axial displacement is u(x) It is loaded by a force q(x) along its length At x = the the displacement u(0) is prescribed to be uˆ At x = L the bar is loaded by a an axial force Nˆ L , positive towards x > The field equations are e = du/d x, N = E Ae, d N /d x + q = 0, and the boundary conditions are u(0) = uˆ and N (L) = Nˆ L (a) Is this problem governed by the Poisson equation, and if so, what is the correspondence with, say, the first of (2.4)? (b) Draw the expanded Tonti diagram for this problem EXERCISE 2.3 [A:25=10+10+5] A steady-state heat conduction problem is posed over the “cylindrical” two-dimensional domain ABC D depicted in Figure E2.1 with dimensions and boundary conditions as shown (Axis x3 comes out of the plane of the paper Domain ABC D extends indefinitely along x3 , and all conditions are independent of that dimension.) The conductivity k is uniform over the domain ABC D x2 q^ = Perfect insulator ^ T =100 D A q^ = Perfect insulator h= B T^ = C x1 10 Figure E2.1 A steady-state heat conduction problem (a) Indicate which portions of the boundary form ST and Sq Can something clever be said about the symmetry plane x1 = that may allow the problem to be posed on only half of the domain? (b) Does the temperature distribution satisfy the Laplace equation ∇ T = 0? (c) Does the “guess solution” T = 100x2 /3 satisfy the field equations and boundary conditions? 2–13 2–14 Chapter 2: DECOMPOSITION OF POISSON PROBLEMS x3 x2 x1 Cross section Figure 2.2 The Saint-Venant torsion problem for Exercise 2.4 EXERCISE 2.4 [A:25] The Saint Venant theory of torsion of a cylindrical bar of arbitrary cross section (cf Figure E2.2) may be posed as follows.4 The problem domain is the bar cross section A, delimited by boundary B This domain is assumed simple connected, i.e the section is not hollow The bar material is isotropic with shear modulus G The primary variable is the two-dimensional stress function φ(x1 , x2 ) This function satisfies the standard Poisson equation (E2.1) ∇ φ = −2Gθ, where θ the torsion angle of rotation (about x3 ) per unit length The function φ must be constant over the boundary B: φ = C In the case of singly-connected cross-section domains (solid bars) this constant can be chosen arbitrarily and for convenience may be taken as zero The applied torque is given by Mt = are A φ d A = G J θ , where J is the torsional rigidity The shear stresses σ13 = ∂φ , ∂ x2 σ23 = − ∂φ ∂ x1 (E2.2) The shear stresses satisfy the equilibrium equations ∂σ13 ∂σ23 + = ∂ x1 ∂ x2 (E2.3) Draw an expanded Tonti diagram for this problem, in which φ is the primary variable, the shear stresses are taken as flux variables, the gradient of φ is the intermediate variable, and angle θ is the source Where would you place the specified-moment condition in the diagram? (Hint: connect it to the source) See, for example, Chapter 11 of Timoshenko and Goodier Theory of Elasticity, McGraw-Hill, 1951 2–14 2–15 Exercises EXERCISE 2.5 [A:10] Draw the expanded Tonti diagram for potential flow (§2.4) Identify governing equations along the strong links in direct form Where in the diagram would you specify a prescribed velocity boundary condition? EXERCISE 2.6 [A:10] Draw the expanded Tonti diagram for electrostatics (§2.5) Identify governing equations along the strong links in direct form 2–15 ... temperature is the primal variable of this formulation If the body is thermally isotropic, the ρ of the previous section becomes the thermal conductivity coefficient k, with a − sign to account for the positive... two centuries The extension of the foregoing equations to an unknown vector function u is straightforward In such a case the first ∇ in (2.1) is the gradient operator and the second ∇ the divergence... §2.2 THE POISSON EQUATION §2.1 INTRODUCTION In the first Chapter it was emphasized that the classical formulation of mathematical models in mechanics and physics leads to the Strong Form (SF) These