Multimode solutions of an ideal plastic flow. A.M. Grundland∗and V. Lamothe† 2012-01-23 2 Abstract 1 0 ThispapercontainsananalysisofmultimodesolutionsexpressibleintermsofRiemanninvariants 2 obtained from complex characteristic elements. A variant of the conditional symmetry method for n constructing this type of solution is proposed. Examples of applications of the proposed approach a toanonstationaryirrotationalidealplasticflowin(2+1)dimensionsarediscussedindetail. Several J newclassesofsolutions, someofthembounded, havebeenobtainedinclosedform. Foraparticular 3 case of a stationary ideal plastic flow, extrusion dies have been drawn. 2 MSC: 35B06 · 35F50 · 35F20 · 75Cxx ] keywords: conditional symmetries; Riemann invariants; multimode solution; ideal plasticity. h p - 1 Introduction h t a Riemann waves represent a very important class of solution of nonlinear first-order systems of PDEs. m They are ubiquitous in the differential equations of mathematical physics, since they appear in all multi- [ dimensional hyperbolic systems and constitute their elementary solutions. Their characteristic feature is 1 that, in most cases, they are expressible only in implicit form. For a homogeneous hyperbolic quasilinear v system of first-order PDEs, 9 9 ∂uα 7 Aiµ(u) =0, i=1,...,p, α=1,...,q, µ=1,...,m, (1) α ∂xi 4 . 1 (where A1,...,Ap) are q×m matrix functions of an unknown function u. Here we adopt the convention 0 that repeated indices are summed unless one of them is in a bracket. 2 1 : ARiemannwavesolutionisdefinedbytheequationu=f(r(x,u)),wheref :R→Rq andthefunction v r(x,u)=λ (u)xi is called the Riemann invariant of the vector λ satisfying the equation ker(cid:0)λ Ai(u)(cid:1)(cid:54)= i i i X 0. These solutions have rank at most equal to one. They are building blocks for constructing more r general types of solutions describing nonlinear superpositions of many waves (k-waves), which are very a interesting from the physical point of view. Until now, the only way to approach this task was through the generalized method of characteristics (GMC) [3, 7, 21, 22, 29, 30] and more recently through the ∗Centre de Recherche Mathématiques, Université du Montréal, C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7,Canada,andDépartementdemathématiquesetinformatiques,UniversitéduQuébec,Trois-Rivières,(QC)G9A5H7, Canada,email:grundlan@crm.umontreal.ca †Département de Mathématiques et Statisque, Université de Montréal, C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C3J7,Canada,email:lamothe@crm.umontreal.ca 1 conditionalsymmetrymethod(CSM)[1,6,13,15,16,28]. TheGMCreliesontreatingRiemanninvariants as new dependent variables (which remain constant along the appropriate characteristic curves of the initialsystem(1)andconstituteasetofinvariantsoftheAbelianalgebraofsomevectorfieldX =ξi(u)∂i a a x with λaξi = 0 for 1 ≤ a ≤ k < p. This leads to the reduction of the dimension of the problem. The i a most important theoretical results obtained with the use of the GMC or CSM [16] include the finding of necessary and sufficient conditions for the existence of Riemann k-waves in multidimensional systems. It was shown [29] that these solutions depend on k arbitrary functions of one variable. Some criteria were also found [29] for determining the elastic or nonelastic character of the superposition between Riemann waves described by hyperbolic systems, which is particularly useful in physical applications. In applications to fluid dynamics and nonlinear field theory, many new and interesting results were obtained [2, 3, 10, 11, 12, 22, 26, 30, 32, 33]. Both the GMC and CSM methods, like all other techniques for solving PDEs, have their limitations. This fact has motivated the authors to search for the means of constructing larger classes of multiple wave solutions expressible in terms of Riemann invariants by allowing the introduction of complex integral elements in the place of real simple integral elements (with which the solutions of hyperbolic systems are built [29]). This idea originated from the work of S. Sobolev [31] in which he solved the wave equation by using the associated complex wave vectors. We are particularly interested in the construction of nonlinear superpositions of elementary rank−k solutions (modes) and the proposed analysis indicates that the language of conditional symmetries is an effective tool for this purpose. This approach is applied to the nonstationary irrotational flow of a ideal plastic material in its elliptic region. The organization of this paper is as follows. Section 2 contains a detailed account of the generalized methodofcharacteristicsforfirst-orderquasilinearsystemsofPDEsinmanydimensionsbasedoncomplex characteristicelements. Insection3weformulatetheproblemofmultimodesolutionsexpressibleinterms ofRiemanninvariantsbymeansofagrouptheoreticalapproach. Thisallowsustoformulatethenecessary and sufficient conditions for constructing these types of solutions. Section 4 presents an application of the developed approach to the system of equations describing a nonstationary irrotational flow of ideal plastic materials in (2+1) dimensions. Several new classes of solutions are obtained in closed form. We obtain several bounded solutions. For the stationary case, limiting ourselves to the region where the gradient catastrophe does not occur, we have drawn extrusion dies and the flow inside the tools. Section 5 summarizes the obtained results and contains some suggestions regarding some further developments. 2 Complex characteristic elements. Themethodologicalapproachassumedinthissectionisbasedonthegeneralizedmethodofcharacteristics whichhasbeenextensivelydeveloped(e.g. in [3,9,14,17,29]andreferencestherein)formultidimensional homogeneous and nonhomogeneous hyperbolic systems of first-order PDEs. A specific feature of that approach is an algebraic and geometric point of view. An algebraization of systems of PDEs was made possible by representing the general integral elements as linear combinations of some special elements associated with those vector fields which generate characteristic curves in the spaces of independent variables X and dependent variables U, respectively. The introduction of these elements (called simple integral elements) proved to be very useful for constructing certain classes of rank-k solutions in closed form. These integral elements proved to correspond to Riemann wave solutions in the case of nonelliptic systems and serve to construct multiple waves (k-waves) as a superposition of several single Riemann waves. 2 The generalized method of characteristics for solving quasilinear hyperbolic first-order systems of can be extended to the case of complex characteristic elements [29]. These elements were introduced not only for elliptic systems but also for hyperbolic systems by allowing the wave vectors to be the complex solutions of the dispersion relation associated with the initial system of equations. Thestartingpointistomakeanalgebraization,accordingto[3,14,29],ofafirst-ordersystemofPDEs (1) in p independent and q dependent variables written in its matrix form Ai(u)u =0, i=1,...,p, i (2) x=(x1,...,xp)∈X ⊆Rp, u=(u1,...,uq)∈U ⊆Rq, whereA1,...,Ap arem×q matrixfunctionsofuandwedenotethepartialderivativesbyuα =∂uα/∂xi. i The matrix Lα satisfying the conditions i uα ∈(cid:8)Lα :AisLα =0, s=1,...,q(cid:9) i i α i at some open given point u ∈ U is called an integral element of the system (2). This matrix L = 0 (cid:0)∂uα/∂xi(cid:1) is a matrix of the tangent mapping du:X →T U given by the formula u X (cid:51)(δxi)→(δuα)∈T U, where δuα =uαδxi. u i The tangent mapping du(x) determines an element of linear space L(X,T U), which can be identified u with the tensor product T U ⊗X∗, (where X∗ is the dual space of X, i.e. the space of linear forms). It u is well known [3, 14, 29] that each element of this tensor product can be represented as a finite sum of simple tensors of the form L=γ⊗λ, where λ ∈ X∗ is a covector and γ = γα ∂ ∈ T U is a tangent vector at the point u ∈ U. Hence, the ∂uα u integral element Lα is called a simple element if rank(Lα) = 1. To determine a simple integral element i i Lα we have to find a vector field γ ∈T U and a covector λ∈X∗ satisfying the so-called wave relation i u (cid:0)λ Ai(u)(cid:1)γα =0. (3) i The necessary and sufficient condition for the existence of a nonzero solution γ for the equation (3) is rank(cid:0)λ Ai(u)(cid:1)=0, <ω(cid:121)γ¯ >=0, ω ∈Φ=An{γ,γ¯}. Here,bytheparenthesis<ω(cid:121)γ >,wedenotethecontractionofthe1-formω ∈T∗U withthevectorfield u γ ∈ T U and similarly for the vector field γ¯ ∈ T U. Multiplying the equations (8) by the 1-form ω ∈ Φ u u we get ξξ¯<ω(cid:121)[γ,γ¯]>λ∧λ¯ =0. (9) 4 Welookforthecompatibilityconditionforwhichthesystem(9)doesnotprovideanyalgebraicconstraints on the coefficients ξ and ξ¯. This postulate means that the profile of the simple modes associated with γ ⊗λ and γ¯⊗λ¯ can be chosen in an arbitrary way for the initial (or boundary) conditions. It follows from (9) that the commutator of the vector fields γ and γ¯ is a linear combination of γ and γ¯, where the coefficients are not necessarily constant, i.e. [γ,γ¯]∈span{γ,γ¯}. So, the Frobenius theorem is satisfied. That is, at every point u of the space of dependent variables U, 0 there exists a tangent surface S spanned by the vector fields γ and γ¯ passing through the point u ∈U. 0 Moreover the above condition implies that there exists a complex-valued function α such that [γ,γ¯]=αγ−α¯γ¯, (10) since [γ,γ¯]=[γ¯,γ]=−[γ,γ¯] holds. Next, from the vector fields γ and γ¯ defined on U-space, we can construct the coframe Ψ, that is the set of 1-forms σ and σ defined on U satisfying the conditions 1 2 <σ (cid:121)γ >=1, <σ (cid:121)γ¯ >=0, σ ,σ ∈Ψ, 1 1 1 2 (11) <σ (cid:121)γ >=0, <σ (cid:121)γ¯ >=1. 2 2 Substituting (7) and (10) into the prolonged system (8) and multiplying by σ and σ respectively, we 1 2 get (i) dξ∧λ+ξ(cid:0)ξ¯λ¯∧λ +ξλ∧λ¯ (cid:1)+αξξ¯λ∧λ¯ =0, ,γ¯ ,γ (12) (ii) dξ¯∧λ¯+ξ¯(cid:0)ξλ∧λ¯ +ξ¯λ¯∧λ (cid:1)+α¯ξξ¯λ¯∧λ=0. ,γ ,γ¯ If dξ = dξ¯ then equation (12.i) is a complex conjugate of (12.ii). So one can consider one of them, say (12.i). We look for the condition of integrability such that the system (12.i) does not impose any restriction on the form of the coefficients ξ and ξ¯. By exterior multiplication of (12.i) by λ, we obtain λ ∧λ¯∧λ=0 (13) ,γ¯ and its respective complex conjugate equation λ¯ ∧λ∧λ¯ =0. (14) ,γ Notethatifconditions(10),(13)and(14)aresatisfied,thenfromtheCartanlemmathesystem(12)and consequently the set of 1-forms (8) has nontrivial solutions for dξ and dξ¯. So under these circumstances the following result holds [29]. Proposition 1 The necessary condition for the Pfaffian system (5) to possess a simple mode solution is that the following two constraints on the complex-valued vector fields γ and λ be satisfied (i) [γ,γ¯]=αγ−α¯γ¯, for any complex-valued function α∈C (ii) λ ,γ¯ ∈span(cid:8)λ,λ¯(cid:9). ,γ¯ ,γ 5 Note that the conditions (10), (13) and (14) are strong requirements on the functions γ =(cid:0)γ1,...,γq(cid:1) and λ = (λ ,...,λ ) and their complex conjugates which satisfy the wave relation (3). Consequently, 1 p the postulated form of a one-mode solution u(x) of the elliptic system (2), required by the generalized method of characteristics (GMC) [21, 29], is such that all first-order derivatives of u(x) with respect to xi are decomposable in the form (5). This restriction is a strong limitation on the admissible class of solutions of (2). So far, there have been few, if any, known examples of such solutions. Therefore, it is worth developing this idea by weakening the integrability conditions (10), (13) and (14) in order to construct multi-mode solutions. In the next section we propose an alternative way of constructing multi-mode solutions (expressed in terms of Riemann invariants) which are obtained from a version of theconditionalsymmetrymethod(CSM)[16]byadaptingittotheellipticsystems. Thisconstitutesthe objective of this paper. 3 Conditional symmetry method and multimode solutions in terms of Riemann Invariants. Inthissection,weexaminecertainaspectsoftheconditionalsymmetrymethodinthecontextofRiemann invariants for which the wave vectors are complex solutions of the dispersion relation (4) associated with the original system (2). Let us consider a nondegenerate first-order quasilinear system of PDEs (2) in its matrix form A1(u)u +...+Ap(u)u =0, (15) 1 p where A1,...,Ap are m×q real-valued matrix functions of u. Let us suppose that there exist k linearly independent non-conjugate complex-valued wave vectors λA(u)=(cid:0)λA(u),...,λA(u)(cid:1)∈Cp, λA (cid:54)=λ¯B ∀ B ∈{1,...,k}, A=1,...,k

t (where we denote the 0 0 independent variables by t = x0,x˜ = x1,...,xn where p = n+1) in which the gradient catastrophe for the solution u(t,x˜) of the system (2) does not take place [30]. Note that the Jacobian matrix of u(x) has at most rank equal to 2k. It follows that the proposed solution (17) is also at most of rank-2k. Its image is a 2k-dimensional submanifold S in the first jet 2k space J1 =J1(X×U). Let us introduce now a set of p−2k linearly independent vectors ξ :Rq →Cp defined by a ξ (u)=(cid:0)ξ1(u),...,ξp(u)(cid:1), a=1,...,p−2k, (27) a a a satisfying the orthogonality conditions λAξi =0, λ¯Aξi =0, A=1,...,k, a=1,...,p−2k, (28) i a i a for a set of 2k linearly independent wave vectors (cid:8)λ1,...,λk,λ¯1,...,λ¯k(cid:9). It should be noted that the set (cid:8)λ1,...,λk,λ¯1,...,λ¯k,ξ1,...,ξp−2k(cid:9) forms a basis for the space of independent variables X. Note also, that the vectors ξ are not uniquely defined since they obey the homogeneous conditions (28). As a a consequence of equation (19) or (20), the graph Γ={x,u(x)} is invariant under the family of first-order differential operators ∂ X =ξi(u) , a=1,...,p−2k, (29) a a ∂xi defined on X ×U space. Since the vector fields X do not include vectors tangent to the direction of u, a they form an Abelian distribution on X×U space, i.e. [X ,X ]=0, a,b=1,...,p−2k. (30) a b Theset(cid:8)r1,...,rk,r¯1,...,r¯k,u1,...uq(cid:9)constitutesacompletesetofinvariantsoftheAbelianalgebra Lgeneratedbythevectorfields(29). Sogeometrically,thecharacterizationoftheproposedsolution(17) of equations (15) can be interpreted in the following way. If u(x) is a q-component function defined on a neighborhood of the origin x=0 such that the graph of the solution Γ={(x,u(x))} is invariant under a setofp−2kvectorfieldsX withtheorthogonalityproperty(28),thenforsomefunctionf theexpression a u(x)isasolutionofequation(17). Hencethegroup-invariantsolutionsofthesystem(15)consistofthose functions u=f(x) which satisfy the overdetermined system composed of the initial system (15) together with the invariance conditions ξiuα =0, i=1,...,p, a=1,...,p−2k, (31) a i ensuring that the characteristics of the vector fields X are equal to zero. a It should be noted that, in general, the conditions (31) are weaker than the differential constraints (5) required by the generalized method of characteristics, since the latter method is subjected to the algebraic conditions (3). In fact, equations (31) imply that there exist complex-valued matrix functions 8 Φα(x,u)andΦα(x,u)definedonthefirstjetspaceJ =J(X×U)suchthatallfirstderivativesofuwith A A respect to xi are decomposable in the following way uα =Φα(x,u)λA+Φα(x,u)λ¯A (32) i A i A i where (cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)∂f ∂f¯(cid:19) Φα = I − + +c.c. + A q ∂r ∂r ∂u ∂r ∂r (33) α (cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)∂f ∂f¯(cid:19) Φ = I − + +c.c. + A q ∂r ∂r ∂u ∂r¯ ∂r¯ or Φα =(cid:18)∂f + ∂f¯(cid:19)M1+(cid:18)∂f + ∂f¯(cid:19)M2 A ∂r ∂r ∂r¯ ∂r¯ (34) Φα =(cid:18)∂f + ∂f¯(cid:19)M1+(cid:18)∂f + ∂f¯(cid:19)M2 A ∂r¯ ∂r¯ ∂r ∂r The matrices ΦαλA and Φαλ¯A appearing in equation (32) do not necessarily satisfy the wave relation A i A i (3). As a result of this fact, the restrictions on the initial data at t = 0 are eased, so we are able to consider more diverse types of modes in the superpositions than in the case of the generalized method of characteristics described in section 2. Let us now proceed to solve the overdetermined system composed of equations (15) and differential constraints (31) Aiµ(u)uα =0, ξi(u)uα =0. (35) α i a i Substituting (19) or (20) into (15) yields the trace condition (cid:32) (cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)(cid:18)∂f ∂f¯(cid:19) (cid:19)(cid:33) tr Aµ I − + +c.c. + λ+c.c. =0 (36) q ∂r ∂r ∂u ∂r ∂r or tr(cid:18)Aµ(cid:20)(cid:18)∂f + ∂f¯(cid:19)(cid:0)M1λ+M2λ¯(cid:1)+c.c(cid:21)(cid:19)=0 (37) ∂r ∂r onthewavevectorsλandλ¯ andonthefunctionsf andf¯,whereA1,...,Aq arep×q matrixfunctionsof u (i.e. Aµ =(Aµi(u))∈Rp×q,µ=1,...,m). For the given initial system of equations (15), the matrices α Aµ areknownfunctionsofuandthetraceconditions(36)or(37)areconditionsonthefunctionsf,f¯,λ, λ¯ (or on ξ due to the orthogonality conditions (28)). From the computational point of view, it is useful to split xi into xiA and xia and to choose a basis for the wave vector λA and λ¯A such that λA =dxiA +λAdxia, λ¯A =dxiA +λ¯Adxia, A=1,...,k, (38) ia ia where (i ,i ) is a permutation of (1,...,p). So, the expressions (23) become A a ∂rA ∂λA ∂r¯A ∂λ¯A = iaxia, = iaxia. (39) ∂uα ∂uα ∂uα ∂uα 9 Substituting (39) into (36) (or (37)) yields tr(cid:18)Aµ(cid:0)I −Q xia(cid:1)−1(cid:18)∂f + ∂f¯(cid:19)Λ(cid:19)=0, µ=1...,m, (40) q a ∂R ∂R or tr(cid:18)Aµ(cid:18)∂f + ∂f¯(cid:19)(cid:0)I −K xia(cid:1)−1Λ(cid:19)=0, µ=1...,m, (41) ∂R ∂R 2k a where R=(r1,...,rk,r¯1,...,r¯k)T and (cid:18)∂f ∂f¯(cid:19)∂λ (cid:18)∂f ∂f¯(cid:19)∂λ¯ (cid:18)∂f ∂f¯(cid:19) Q = + ia + + ia = + η ∈Cq×q, a ∂r ∂r ∂u ∂r¯ ∂r¯ ∂u ∂R ∂R a  ∂λia (cid:16)∂f + ∂f¯(cid:17) ∂λia (cid:16)∂f + ∂f¯(cid:17)  (cid:18)∂f ∂f¯(cid:19) (42) Ka = ∂∂λ¯uia (cid:16)∂∂fr + ∂∂fr¯(cid:17) ∂∂λ¯uia (cid:16)∂∂fr¯ + ∂∂fr¯¯(cid:17) =ηa ∂R + ∂R ∈C2k×2k, ∂u ∂r ∂r ∂u ∂r¯ ∂r¯ and for simplicity of notation, we note ∂λ  (cid:32)λ(cid:33) ∂uia ∂f (cid:18)∂f ∂f(cid:19) Λ= ∈C2k×p, η = ∈C2k×q, = , ∈Cq×2k, (43) λ¯ a ∂λ¯  ∂R ∂r ∂r¯ ia ∂u fori fixedandi =1,...,p−1. In(42)the2k×2k matrixK isdefinedintermsofthek×k subblocks A a a of the form ∂λ /∂u(cid:0)∂f/∂r+∂f¯/∂r(cid:1), where η is a matrix form of the block ∂λ /∂u over the block ∂λ¯ /∂u. Thenoiatation(∂f/∂r,∂f/∂r¯)representsathematrixformedoftheleftblocka∂f/∂r andtheright a block ∂f/∂r¯. Note that the functions rA, r¯A and xia are functionally independent in a neighborhood of x = 0 and the matrix functions Aµ, ∂f/∂r, ∂f/∂r¯, ∂f¯/∂r, ∂f¯/∂r¯, Q and K depend on r and r¯only. a a So, equations (40) (or (41)) have to be satisfied for any value of coordinates xia. As a consequence, we have some constraints on these matrix functions. From the Cayley-Hamilton theorem, we know that for any n×n invertible matrix M, the expression (M−1detM) is a polynomial in M of order (n−1). Thus, using the tracelessness of the expression Aµ(cid:0)Iq−Qaxia(cid:1)−1(∂f/∂R)Λ, we can replace equations (40) by the following condition (cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19) tr AµQ + Λ =0, where Q=adj(I −Q xia)∈Cq×q, (44) ∂R ∂R q a where adjM denotes the adjoint of the matrix M. As a consequence the matrix Q is a polynomial of order (q −1) in xia. Taking (44) and all its partial derivatives with respect to xia (with r, r¯ fixed at x=0), we obtain the following conditions for the matrix functions f(r,r¯) and λ(f(r,r¯)) (cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19) tr Aµ + Λ =0, µ=1,...,m, (45) ∂R ∂R (cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19) tr AµQ ...Q + Λ =0, (46) (a1 as) ∂R ∂R 10

Multimode solutions of first-order quasilinear systems obtained from Riemann invariants. Part I

Similar Multimode solutions of first-order quasilinear systems obtained from Riemann invariants. Part I