Lagrangian cell-centered conservative scheme

Transcription

1 Appl. Math. Mech. -Engl. E., 33(10), (2012) DOI /s c Shanghai University an Springer-Verlag Berlin Heielberg 2012 Applie Mathematics an Mechanics (English Eition) Lagrangian cell-centere conservative scheme uan-wen GE ( ) (Institute of Applie Physics an omputational Mathematics, Beijing , P. R. hina) Abstract This paper presents a Lagrangian cell-centere conservative gas ynamics scheme. The piecewise constant pressures of cells arising from the current time sub-cell ensities an the current time isentropic spee of soun are introuce. Multipling the initial cell ensity by the initial sub-cell volumes obtains the sub-cell Lagrangian masses, an iviing the masses by the current time sub-cell volumes gets the current time subcell ensities. By the current time piecewise constant pressures of cells, a scheme that conserves the momentum an total energy is constructe. The vertex velocities an the numerical fluxes through the cell interfaces are compute in a consistent manner ue to an original solver locate at the noes. The numerical tests are presente, which are representative for compressible flows an emonstrate the robustness an accuracy of the Lagrangian cell-centere conservative scheme. Key wors sub-cell force, Lagrange cell-centere scheme, Lagrangian cell-centere conservative gas ynamics scheme, piecewise constant pressure of cell hinese Library lassification O Mathematics Subject lassification 76N15, 65M06 1 Introuction We are intereste in solving the gas ynamics equation written in the Lagrangian form. In the present paper, we aim at presenting a Lagrangian cell-centere conservative gas ynamics scheme. This scheme is the extension of the cell-centere Lagrangian scheme for twoimensional compressible flow problem presente in Ref. [1]. Before escribing the metho, we briefly give a historical overview on the Lagrangian schemes. In Lagrangian hyroynamics methos, a computational cell moves at the flow velocity. In practice, this means that the cell vertices move at a compute velocity, an the cell faces are uniquely specifie by the vertex positions. This ensures that there is no mass flux crossing the bounary of the Lagrangian moving cell. Thus, Lagrangian methos can capture contact iscontinuity sharply in multimaterial flui flows. However, in the Lagrangian framework, one has to iscretize not only the gas ynamics equation but also the vertex motion in orer to move the mesh. Moreover, the numerical fluxes of the physical conservation laws must be etermine in a compatible way with the vertex velocity such that the geometric conservation law (GL) can be satisfie, namely, the rate of change of a Lagrangian volume has to be compute coherently with the Receive Aug. 29, 2011 / Revise May 2, 2012 Project supporte by the National Natural Science Founation of hina (No ) orresponing author uan-wen GE, Associate Professor, Ph.D., ge quanwen@iapcm.ac.cn

2 1330 uan-wen GE noe motion. This critical requirement is the cornerstone of any Lagrangian multi-imensional scheme. The most natural way to solve this problem is to use the staggere iscretization. In the staggere iscretization, the position, the velocity, an the kinetic energy are centere at the points, while the ensity, the pressure, an the internal energy are within the cells. The issipation of the kinetic energy into the internal energy through shock waves is ensure by an artificial viscosity term. Since the seminal works of von Neumann an Richtmyer [2] an Wilkins [3], a lot of evelopments have been mae to improve the accuracy an the robustness of the staggere hyroynamics [4 6]. More specifically, the construction of the compatible staggere iscretization leas to a scheme that conserves the total energy in a rigorous manner [7 8]. We also note the recent evelopment of a variational multi-scale stabilize approach in the finite element computation of Lagrangian hyroynamics piecewise linear approximation for the variables [9 10]. The case of 1 /P 0 finite element was stuie in Ref. [11], where the kinematic variables were represente using a piecewise linear continuous approximation, an the thermoynamic variables utilize a piecewise constant representation. An alternative option to the previous iscretizations is to erive a Lagrangian scheme base on the Gounov metho [12]. In comparison with staggere iscretizations, the Gounov-type methos exhibit the goo properties, i.e., they are naturally conservative, they o not nee an artificial viscosity, an they allow a straightforwar implementation of conservative remapping methos when they are use in the context of the arbitrary Lagrangian-Eulerian (ALE) strategy. In the Gounov-type methos, all conserve quantities incluing momentum an cell velocities are cell-centere. The cell-face quantities incluing a face-normal component of the velocity are available from the solution to an approximate Riemann problem at each cell face. However, the vertex velocity must be etermine to move the mesh. Aessio et al. [13] use a weighte least squares algorithm to compute the vertex velocity on the conition that the vertex velocity projecte in the normal irection of a face is equal to the Riemann velocity on that face. It turns out that this algorithm is capable of generating aitional spurious components in the vertex velocity fiel. Hence, it leas to the artificial gri motion which requires a very expensive treatment [14]. This rawback comes probably from the fact that the flux computation is not compatible with the noe isplacement, an hence the GL is not satisfie. An important achievement concerning the compatibility between the flux iscretization an the vertex velocity computation was introuce by Despŕes an Mazeran [15]. They presente a scheme in which the interface fluxes an the noe velocity were compute coherently with an approximate Riemann solver locate at the noes. This original approach leas to a first-orer conservative scheme which satisfies a local semiiscrete entropy inequality. The multi-imensional higher-orer extension of this scheme was evelope in Ref. [16]. A thorough stuy [1] of the properties of the Despŕes- Mazeran [15] noal solver showe a strong sensitivity to the cell aspect ratio, which coul lea to severe numerical instabilities. To overcome this ifficulty, Maire et al. [1] propose an alternative scheme that successfully solve the aspect ratio problem an kept the compatibility between flux iscretization an vertex computation. The main feature lay in the iscretization of the pressure graient, which meant that two pressures were set at each noe of a cell, an the pressure of each noal was associate with the irection of the outwar unit normal vectors relate to the eges originating from the noe. These noal pressures were linke to the noal velocity by the half Riemann problems. However, Maire et al. [1] faile to remove the numerical ifficulties of anomalous gri istortion that have plague the Lagrangian schemes because of their inception. The bane of Lagrangian hyroynamics calculations is the premature breakown of the gri topology that results in severe egraation of accuracy an run termination before the assumption of a Lagrangian zonal mass is prove to be vali. At short spatial gri scales, this is usually calle the hourglass moe or the keystone motion associate, in particular, with constraine gris such as quarilaterals an hexaherons in two an three imensions, respectively. At longer spatial lengths relative to the gri spacing, there is the spurious vorticity

3 Lagrangian cell-centere conservative scheme 1331 or the long-thin zone problem. In both cases, the result is anomalous gri istortion an tangling that has nothing to o with the actual solution. In this work, we show how such motion can be eliminate by the proper usage of piecewise constant pressures of cells. These piecewise constant pressures of cells arise from the current time isentropic spee of soun of cells an the current time sub-cell ensities. Multipling the initial cell ensity by the initial sub-cell volumes obtains the sub-cell Lagrangian masses, an iviing the sub-cell Lagrangian masses by the current time sub-cell volumes gets the current time sub-cell ensities. This paper is organize as follows. The cell-centere Lagrangian scheme for two-imensional compressible flows is introuce in Section 2. We also introuce the concept of sub-cell forces borrowe from the staggere iscretization framework [7] to erive a general form of the cell-centere iscretization. The Lagrangian cell-centere conservative gas ynamics scheme is introuce in Section 3. Extensive numerical experiments are reporte in Section 4. The experiments show not only the robustness an accuracy of the present metho but also the ability to successfully hanle complex two-imensional flows. onclusions an perspectives are given in Section 5. 2 Notation an previous work 2.1 Lagrangian hyroynamics Let D be an open subset of R 2, which is fille with an invisci ieal flui an equippe with the orthonormal frame (O, X, Y ) an the orthonormal basis (e X, e Y ). Define the unit vector e Z = e X e Y. Discretize the equations of Lagrangian hyroynamics as follows: V (t) V (t) V (t) V (t) ρ V = 0, (1) V U NS = 0, (2) S(t) ρ UV + PNS = 0, (3) S(t) ρev + PU NS = 0, (4) S(t) where enotes the Lagrangian time erivation. V (t) is the moving control volume, an S(t) is the bounary of V (t). ρ, U = (u, v) T,, an E are the specific volume, the velocity, the pressure, an the specific total energy of the flui, respectively. N enotes the outwar unit normal vector relate to the moving bounary S(t). Equations (1), (3), an (4) express the conservation of mass, momentum, an total energy, respectively. We note that Eq.(2) is also name the GL, an it is equivalent to the following local kinematic equation: X = U, X(0) = x, (5) where X stans for the coorinates efining the control volume surface at time t > 0, an x stans for the coorinates at time t = 0. Then, X = X(x, t) is implicitly efine by the local kinematic equation, which is also calle the trajectory equation. This enables us to efine the map M t : V (0) V (t), X X(x, t), where X is the unique solution to (5). With fixe t, this map transforms each flui particle from the position at time t = 0 to the position at time t. The thermoynamical closure of the

4 1332 uan-wen GE set of Eqs.(1), (3), an (4) is obtaine by the aitional equation of state in the form of P = P(ρ, ε), (6) where the specific internal energy ε is relate to the specific total energy by ε = E 1 2 U Notation an assumptions Let us consier the physical omain V (0) initially fille with the flui. We assume that we can map it by a set of quarilateral cells Ω (0) ( = 1, 2,, I) without gaps or overlaps, where I is the total number of the cells. Each cell is assigne a unique inex an enote by Ω (0). Using the map M t, we set Ω (t)=m t (Ω (0)). Here, we assume that Ω (t) is still a quarilateral, that is, the map M t is a continuous an linear function over each element of the mesh. Each point (vertex) of the mesh Ω (t) is assigne a unique inex, an we enote by () the counter clockwise orere list of points of cells. The vertices of cells Ω (t) are perioic, i.e., ()+1 = 1, 0 = (). () is the set of the cells aroun point (vertex), an the numbering is also perioic, i.e., Ω ()+1 (t) = Ω 1 (t). We use inex f to enote a generic face of cell Ω (t). L f is the length, Nf is the outwar unit normal vector relate to the face, an F is the set of faces of cells (see Fig. 1). We also introuce the face fluxes Uf, Π f, an (ΠU) f as follows: Uf = 1 L US, f f Π f = 1 L PS, f f (ΠU) f = 1 PUS. L f f (7) Fig. 1 Notation relate to quarilateral cell Ω (t) To get the iscrete evolution equations for the primary variables (τ, U, E), the control volume formulation is use to the quarilateral cell Ω (t). Let m enote the mass of cell Ω (t), which is a constant after Eq.(1). For a flow variable φ, we introuce its mass average value over the cell Ω (t), i.e., φ = 1 ρφv. m Ω (t)

5 Lagrangian cell-centere conservative scheme 1333 Then, Eqs. (1), (3), an (4) are τ m U m + E m + f F() f F() f F() L f U f N f = 0, L f Π f N f = 0, L f (ΠU) f N f = 0. (8) The local kinematic equations in a iscrete form at the point are X = U, X (0) = x, where X enotes the coorinates of point at time t > 0, x is the initial position, an U is the velocity of point. Equation (8) represents the face flux iscretization of the Lagrangian hyroynamics equations for the iscrete variables. In orer to compute the time evolution of the flow variables, we nee to calculate the face fluxes Uf, Π f, an (ΠU) f. Moreover, we also nee to compute the point velocity U to move the mesh. 2.3 ompatible iscretization of GL Since m /ρ = V, m, ρ, an V are the mass, the ensity, an the volume of cell Ω (t), respectively. Equation (2) can be rewritten as follows: V f F() L f U f N f = 0. (9) V is the volume of cell Ω (t), which is a function of coorinates X of point for (). We compute this volume by performing the triangular ecomposition of cell Ω (t) as follows: which is shown in Fig.2. V = 1 2 () (X X +) e Z, Fig. 2 Triangular ecomposition of quarilateral cell Ω (t) The time ifferentiation of this equation leas to V = () 1 2 (L N + L +N +) U, (10) where L an L + are the lengths, an N an N + are the outwar unit normal vectors relate to the eges an +. By shifting inices in the previous summation,

6 1334 uan-wen GE Eq.(10) becomes V = () 1 2 L +N + (U + U +). (11) Now, the comparison between Eqs.(9) an (11) shows that they are equivalent uner the conition U f = 1 2 (U + U +). (12) The only way to satisfy the compatibility conition (12) is to first compute the point velocity U an euce the face velocity Uf. Then, the compatibility of the face iscretization of the GL with the rate of change of the cell volume can be ensure. Introuce notation L = 1 2 L, N = N, L = 1 2 L +, N = N +. Equation (10) can be rewritten as V = () L N U, where N stans for the unit corner vector relate to point an cell Ω (t) efine by L N = L N + L N. 2.4 omputation of momentum flux Maire et al. [1] iscretize the momentum flux by introucing two pressures at each noe of cell Ω (t). These pressures are enote by Π an Π, as shown in Fig.3. They can be seen as noal pressures viewe from cell Ω (t) an relate to the two eges impinging at noe. Using these noal pressures, Maire et al. [1] propose the following efinition of the iscrete graient operator ( P) over cell Ω (t): ( P) = 1 V () (L Π N + L Π N ). Fig. 3 Localization of noal pressures given by half Riemann problems at point Using the iscrete graient operator, the momentum equation is rewritten as m U + () (L Π N + L Π N ) = 0. (13)

7 Lagrangian cell-centere conservative scheme 1335 Maire et al. [1] obtaine a noal flux iscretization for the momentum equation which is equivalent to its face flux iscretization (11) provie that the momentum face flux is written as Π f = 1 2 (Π + Π ). Introuce force F, which is a sub-cell force relate to point an cell Ω (t), Then, the momentum equation can be written as F = L Π N + L Π N. (14) m U + () F = 0. (15) Since the velocity of eges an + in the vicinity of point is equal to the noal velocity, the noal pressures Π an Π can be compute using the following half approximate Riemann problems: P Π = Z (U U ) N, P Π = Z(U U ) N. (16) Here, Z an Z are the mass fluxes swept by the waves. To etermine the coefficients, we follow the approach by setting Z = ρ a + ρ Γ (U U ) N, Z = ρ a + ρ Γ (U U ) N, (17) where a is the local isentropic spee of soun, an Γ is a material-epenent parameter that is given in terms of the ensity ratio in the limit of very strong shocks. In the case of Gamma law gas, one gets Γ = (γ +1)/2. Utilizing Eq.(14), the sub-cell force F can be rewritten as F = L P N M (U U ), (18) where N stans for the unit corner vector relate to point an cell Ω (t) efine by L N = L N + L N, an M = Z L (N N ) + Z L (N N ) is a 2 2 symmetric positive efinite matrix. 2.5 omputation of total energy flux The time rate of change of total energy is equal to the summation of the work performe by the sub-cell forces F over cell Ω (t). The substitution of the sub-cell force efinition of F in the previous equation leas to the following noe flux iscretization of the total energy equation: m E + () (L Π N + L Π N ) U = 0. (19) 2.6 onservation of momentum an total energy To examine the momentum conservation, let us write the global balance of momentum without taking the bounary conitions into account. The summation of the momentum equation (13) over all the cells leas to ( ) m U = F. (20) ()

8 1336 uan-wen GE Switching the summation over cells an the summation over noes in the right-han sie of Eq.(20) yiels ( ) m U = F, (21) () where () is the set of cells aroun point. The summation of the total energy equation (19) over all the cells yiels ( ) m E = F U. (22) () Switching again the summation over cells an the summation over noes in the right-han sie of Eq. (22) yiels ( ) m E = F U. (23) () Then, the total energy conservation is ensure provie that the sub-cell forces satisfy the following conition: F = 0. (24) () 2.7 onstruction of noal solver By Eq.(18), the sufficient conition (24) can be rewritten as Set () (L P N M (U U )) = 0. (25) M = () M. Then, the system satisfie by the point velocity U is written as M U = () (L P N + M U ). (26) 2.8 Problem statement V = (L N + L N) U, () U m + (L Π N + L Π N) = 0, () E m + (L Π N + L Π N) U = 0, M U = () (L P N + M U ), () P Π = Z (U U ) N, P Π = Z(U U ) N. (27)

9 Lagrangian cell-centere conservative scheme 1337 Maire et al. [1] claime that, when conition (24) is satisfie, the momentum an the total energy are conserve. However, we fin that conition (24) is incorrect, because two pressures are set at each noe of cell Ω (t), an the pressure iscontinuous line exists insie cell Ω (t), as shown in Fig.4. Similarly, the pressure iscontinuous line ( = 1, 2,, ()) exists insie each cell Ω (t) ( = 1, 2,, ()). Due to the presence of these pressure iscontinuous lines ( = 1, 2,, ()), conition (24) is incorrect. Our purpose is to present a Lagrangian cell-centere conservative gas ynamics scheme. Instea of following the proceure employe by Maire et al. [1], we fin it more transparent an physically motivate to utilize a piecewise constant pressure of cell Ω (t). We next efine the piecewise constant pressure of cell Ω (t). In Fig.5, we show a quarilateral cell labelle Ω (t), single out one of its efining points (vertices), an label it as. We efine the sie mipoints of a given cell Ω (t) an label them as B 1, B 2, B 3, an B 4. Using the cell points (vertices), +, k, an an the sie mipoints B 1, B 2, B 3, an B 4 of a given cell Ω (t), we subivie a quarilateral Ω (t) into eight triangle sub-cells B 1, B 2, + B 2, + B 3, k B 3, k B 4, B 4, an B 1. onsiering the triangle sub-cells B 1, B 2, + B 2, + B 3, k B 3, k B 4, B 4, an B 1, we enote the pressures by Π, Π, Π, Π, Π, Π, Π, an Π. Then, the pressure of + + k k cell Ω (t) is enote as the piecewise constant pressures Π, Π, Π +, Π +, Π k, Π k, Π, an Π in triangle sub-cells B 1, B 2, + B 2, + B 3, k B 3, k B 4, B 4, an B 1. onsiering the triangle sub-cell B 2, we enote its volume, pressure, mass, an ensity by Ṽ, Π, m, an ρ, respectively. onsiering the triangle sub-cell B 1, we enote its volume, pressure, mass, an ensity by Ṽ, Π, m, an ρ, respectively. The forces ue to these piecewise constant pressures can be calculate along the bounaries of these sub-cells. Then, we introuce force F, which is a sub-cell force relate to point an cell Ω (t), i.e., where F = L Π N + L Π N + D Π + D Π, (28) Π = a2 ( ρ ρ ) + P, Π = a2 ( ρ ρ ) + P, (29) in which a is the isentropic spee of soun of cell Ω (t), the length an the outwar unit normal vector D are relate to ege of triangle sub-cell B 1, an the length an the outwar unit normal vector D are relate to ege of the triangle sub-cell B 2. Fig. 4 Notation relate to noal solver at point of cell Ω (t) Fig. 5 Localization of piecewise pressure at point viewe from cell Ω (t)

10 1338 uan-wen GE 3 onservative spatial approximation 3.1 omputation of momentum flux Using the force, we propose the following efinition of the iscrete graient operator over cell Ω (t): ( P) = 1 V () Then, the momentum equation can be written as m U m U + () + () (L Π N + L Π N + D Π + D Π ). (30) (L Π N + L Π N + D Π + D Π ) = 0. (31) F = 0. (32) 3.2 omputation of total energy flux The time rate of change of total energy is equal to the summation of the work performe by the sub-cell forces F over cell Ω (t), i.e., m E + () F U = 0. (33) Substituting the sub-cell force F in Eq.(33) yiels the noe flux iscretization of the total energy equation, i.e., m E + () (L Π N + L Π N + D Π + D Π ) U = 0. (34) 3.3 Noe flux iscretization for cell Ω (t) Gathering the results from the previous sections, we write the semi-iscrete Lagrangian cell-centere conservative gas ynamics scheme of noe flux iscretization for the unknowns (τ, U, E ) as follows: τ m = (L N + L N) U, () U m + (L Π N + L Π N + D Π + D Π ) = 0, (35) () E m + (L Π N + L Π N + D Π + D Π ) U = 0. () 3.4 onservation of momentum an total energy To examine the momentum conservation, let us write the global balance of momentum without taking the bounary conitions into account. The summation of the momentum equation (32) over all the cells leas to ( ) m U = () F. (36)

11 Lagrangian cell-centere conservative scheme 1339 Switching the summation over cells an the summation over noes in the right-han sie of Eq.(36) yiels ( ) m U = () F. (37) Then, the momentum conservation is ensure provie that the sub-cell forces satisfy the following conition: () F = 0. (38) Since the sum of all forces acting on one point is equal to zero (see Fig. 6), conition (38) is satisfie, an the momentum is conserve. We claim that the total energy is also conserve. To emonstrate this property, we perform the summation of the total energy equation (33) over all the cells, ( ) m E = () F U. (39) Then, we switch again the summation over cells an the summation over noes in the right-han sie of Eq.(39) to obtain ( ) m E = () ( () F ) U. (40) F = 0. (41) Due to Eq.(38), the term between parentheses in the right-han sie of Eq.(40) is null, an the total energy is conserve. Fig. 6 Notation relate to noal solver at point 3.5 onstruction of solver at vertex velocity U We can introuce only one pressure at point. This pressure is etermine by the half Riemann problem efine in the irection of the unit corner vector N, i.e., P Π = Z (U U ) N, (42)

12 1340 uan-wen GE where Z is the acoustic impeance of cell Ω (t), Z = ρ a + ρ Γ (U U ) N. (43) This amounts to efine only one noal pressure Π for each cell Ω (t) that surrouns point, as shown in Fig. 7. The sub-cell force F corresponing to the single noal pressure Π is F = L P N M (U U ), (44) where N stans for the unit corner vector relate to point an cell Ω (t) efine by L N = L N + L N, an M = Z L N N is a 2 2 symmetric positive efinite matrix. Since the sum of all forces acting on point is equal to zero, we get () (L P N M (U U )) = 0. (45) Using conitions (28), (29), an (38), we write the linear system satisfie by the components (u, v ) of velocity U of an internal vertex. This generic vertex is not on the bounary of the omain so that it is surroune by cells Ω (t)( = 1, 2,, ()), as shown in Fig. 7. () (L a2 ( ρ ρ )N + L a2 ( ρ ρ )N + D a2 ( ρ ρ ) () + D a2 ( ρ ρ )) + (L P N + L P N ) = 0. (46) Fig. 7 Notation of one pressure Π at point of cell Ω (t) Substituting Eq.(45) to Eq.(46), we can obtain the following equation satisfying the point velocity U : () (L a 2 ( ρ ρ )N + L a2 ( ρ ρ )N + D a 2 ( ρ ρ ) () + D a2 ( ρ ρ )) + ( M (U U )) = 0. (47)

13 Set M = () () M U = Lagrangian cell-centere conservative scheme 1341 M. Then, the system satisfie by the point velocity U is written as () M U (L a2 ( ρ ρ )N + L a2 ( ρ ρ )N + D a2 ( ρ ρ ) + D a2 ( ρ ρ )). (48) Denote (u, v ) by the components of velocity U of an internal vertex satisfying { A u + v = S X, u + B v = S Y, (49) where A, B, an are efine by A = B = = () () () Z (L (N,X )2 + L (N,X )2 ), Z (L (N,Y )2 + L (N,Y )2 ), Z (L (N,X )(N,Y ) + L (N,X )(N,Y )). The right-han sie of Eq.(49) is the components (S X, S Y ) of vector Š efine by Š = () () M U (L a2 ( ρ ρ )N + L a2 ( ρ ρ )N + D a2 ( ρ ρ ) + D a2 ( ρ ρ )). The eterminant of Eq.(49) is = A B 2. We show that it is always positive. In orer to simplify the notation, we efine ( W X = Z1 L 1 N1, ) Z 1 L 1,X N1,, Z () L () N (), Z () L (),X,X N (),,X ( W Y = Z1 L 1 N1, ) Z 1 L 1,Y N1,, Z () L () N (), Z () L (),Y,Y N ().,Y We can immeiately obtain A = W X 2, B = W Y 2, an = W X, W Y, where, is the inner scalar prouct of R 2, an is its associate norm. From the auchy-schwarz inequality, we know that A B 2 0. In fact, A B 2 = 0 if an only if one of the two vectors is null or W X an W Y are colinear. This situation is generally impossible unless the eges aroun noe merge into a single line. The system (49) has always a unique solution which etermines the velocity (u, v ). 3.6 Bounary conitions In this paragraph, we explain our implementation of bounary conitions which is consistent with our internal solver. In the Lagrangian formalism, we have to consier two types of bounary conitions, the pressure or the normal component of the velocity. We use the same

14 1342 uan-wen GE type of notation as in Subsection 3.5. is on the bounary. It is surroune by () cells containe in the omain. There are () + 1 eges impinging on. They are numbere counter clockwise, as shown in Fig. 8. The first Ω 1 an the last cells Ω () have eges M 1 an M ()+1 on the bounary. The outwar normal vectors to the two bounary eges M 1 an M ()+1 coming out of are enote by N 1 respectively. an N() coherently with our notations, Fig. 8 Notation for bounary conitions ase1 A prescribe pressure We enote by Π 1 an Π ()+1 the pressures that are impose on the bounary eges M 1 an M ()+1 (see Fig. 8), respectively. For the conservation relation, we make a balance aroun the vertex ue to the bounary conitions an the conitions (45). We get () Set M = (L P N M (U U )) = L 1 N1 Π 1 + L () () M U = N () Π ()+1. (50) M. Then, the system satisfie by the point velocity U is written as () The components of U satisfy () M U + L P N L 1 N1 Π 1 L() N () Π ()+1. (51) { A u + v = S X, u + B v = S Y. (52) The coefficients A, B, an have alreay been efine in the previous paragraph. The right-han sie of Eq. (52) is the components ( S X, S Y ) of vector S efine by S = () () M U + L P N L 1 N1 Π 1 L () N () Π ()+1. The eterminant of Eq. (52) is = A B 2. It is positive, an system (52) always has a unique solution which etermines the velocity (u, v ). ase2 A prescribe normal velocity Let W 1 an W ()+1 be the values of the prescribe normal velocities on the bounary eges M 1 an M ()+1 coming out of. We istinguish the following two cases. (i) N 1 an N() are not colinear.

15 Lagrangian cell-centere conservative scheme 1343 In this case, the value of the point velocity U is given by the bounary conitions, an the components (u, v ) of the vertex velocity are the solution to the following linear system: N 1,X u + N1,Y v = W 1, N (),X u + N(),Y v = W ()+1. This linear system (53) always has a unique solution. Since the normal vectors are not colinear, the eterminant is not zero. (ii) N 1 an N() are colinear. In this case, the point velocity U is not given irectly, an we nee to know the balance of momentum aroun vertex taking the bounary conitions into account. We get () Set M = (L P N M (U U )) = (L 1 N1 + L() () M U = N () (53) )Π. (54) M. Then, the system satisfie by the point velocity U is written as () () M U + L P N (L 1 N1 + L() N () )Π, (55) where Π is an average pressure applie on the external sie of eges M 1 an M ()+1 coming out of. The pressure is unknown. However, we have an aitional equation corresponing to the bounary conition, i.e., (L 1 N1 + L() N () ) U = L 1 W 1 + L () W ()+1. (56) ombining Eq.(55) with Eq. (56), we get A u + v + GΠ = ŜX, u + B v + HΠ = ŜY, Gu + Hv = L1 W 1 + L () W ()+1, (57) where ŜX an ŜY are the components of vector Ŝ efine by Ŝ = () () M U + L P N. The coefficients A, B, an have alreay been efine in the previous paragraph, an we set G = L 1 N1 +,X L() N () an H =,X L1 N1 +,Y L() N (). The eterminant of (57),Y is = A H GH B G 2. Using the fact that A B 2 > 0, we can show that < 0, an Eq. (57) always has a unique solution (u, v ). 3.7 Time iscretization an mesh motion In this section, we iscretize the system with respect to time that escribes the evolution of the physical variables (τ, U, E ) in cell Ω (t). We compute the volume, momentum, an total energy fluxes Uf N f, Π f N f, an (ΠU) f N f on face + by the volume, momentum,

16 1344 uan-wen GE an total energy fluxes on points an + as follows: Uf N f = 1 2 (U + U +) Nf, L f Π f N f = 1 2 ((L Π N + L Π N + D Π + D Π ) +(L Π N + L Π N + D Π D Π )), + + L f (ΠU) f N f = 1 2 ((L Π N + L Π N + D Π + D Π ) U +(L Π N + L Π N + D Π D Π ) U + + +), where f enotes face + of cell Ω (t). Then, system (35) is written as τ m = 1 f F() 2 L f (U + U +) Nf, U m + 1 f F() 2 ((L Π N + L Π N + D Π + D Π ) +(L Π N + L Π N + D Π D Π )) = 0, + + E m + 1 f F() 2 ((L Π N + L Π N + D Π + D Π ) U +(L Π N + L Π N + D Π D Π ) U + + +) = 0. (58) For the time iscretization of Eq.(58), we use a classical forwar Euler scheme. We assume that the physical properties U,n, E,n, ρ,n, a,n, an P,n in Ω (t) an the geometrical characteristics X,n, Y,n ( = 1, 2,, ()) of cell at the beginning of the time step, i.e., t n, are known. We are going to compute their values at t n+1, an we set t = t n+1 t n. The noal solver (49) an system (52), (53), an (57) provie the vertex velocities (u, v ) from the physical variables an geometry characteristics evaluate at time t n. The explicit time integration of the trajectory equation provies the location of vertices for any time t [t n, t n+1 ], { X (t) = X,n + (t t n )u, Y (t) = Y,n + (t t n )v. (59) By this way, we get the location of the vertices at time t n+1, { X,n+1 = X,n + tu, Y,n+1 = Y,n + tv. (60) Then, we euce that L +N +(t) = ((Y +(t) Y (t)), (X +(t) X (t))) T is linear in time. Thus, tn+1 t n (L +N +)(t) = t 2 ((L +N +) n + (L +N +) n+1 ). The last results enable us to write an approximation of the volume equation which is coherent with the mesh motion, namely, m (τ,n+1 τ,n ) t 4 F() ((L f Nf ) n + (L f Nf ) n+1 )(U + U +) = 0. (61) f=1

17 Lagrangian cell-centere conservative scheme 1345 The approximation of the momentum an the total energy equation is fully explicit to conserve the momentum an the total energy exactly. The lengths of eges an the normal vectors have the same efinition as previously. We get m (U,n+1 U,n ) + t f F() 1 2 ((L Π N + L Π N + D Π + D Π ) +(L Π + +N + + L Π + +N + + +D Π D Π + +)) = 0, (62) an for the total energy, m (E,n+1 E,n ) + t f F() 1 2 ((L Π N + L Π N + D Π + D Π ) U +(L Π + +N + + L Π + +N + + +D Π D Π + +) U +) = 0. (63) The time step t is evaluate following criterion. At t n, for each cell Ω (t), we enote by λ,n the minimal value of the istance between two noes of cell Ω (t). We efine t = λ,n min, (64),2,,I ã,n where ã,n is the maximal value of the local isentropic spees of soun of triangle sub-cells an cell Ω (t). 3.8 Description of algorithm (i) Initialization We know ensity ρ 0 an coorinate of vertices (X0, Y 0 ) ( = 1, 2,, ()) of cells Ω (t) ( = 1, 2,, I) at the initial time, where I is the total number of cells. We compute the triangle sub-cell masses m, m, m, m, m, m, m an m from the initial + + k k ensity ρ 0 an the coorinate of vertices (X0, Y 0 ) ( = 1, 2,, ()) of cells Ω (t) ( = 1, 2,, I). At time t n, in each cell Ω (t) ( = 1, 2,, I), we know the flui variables U,n, E,n, ρ,n, a,n, an P,n an the geometrical characteristics (X,n, Y,n ) ( = 1, 2,, ()). We compute the triangle sub-cell volumes (Ṽ ) n, (Ṽ ) n, (Ṽ + ) n, (Ṽ + ) n, (Ṽ k ) n, (Ṽ k ) n, (Ṽ ) n, an (Ṽ ) n an ensities of triangle sub-cells ( ρ ) n, ( ρ ) n, ( ρ + ) n, ( ρ + ) n, ( ρ k ) n, ( ρ k ) n, ( ρ ) n, an ( ρ ) n. Using pressure P,n an ensities of triangle sub-cell ( ρ ) n, ( ρ ) n, ( ρ + ) n, ( ρ + ) n, ( ρ k ) n, ( ρ k ) n, ( ρ ) n, an ( ρ ) n, we get the isentropic spee of soun of triangle sub-cells (ã ) n, (ã ) n, (ã + ) n, (ã + ) n, (ã k ) n, (ã k ) n, (ã ) n, an (ã ) n. We compute the pressures of triangle sub-cells ( Π ) n, ( Π ) n an the sub-cell force ( F ) n by solving Eqs.(28) an (29). (ii) Noal solver For each internal vertex, we compute velocity (u, v ) by solving the linear system (49), n an for each bounary vertex, we compute velocity (u, v ) by solving the linear system (52), n (53), an (57). (iii) Time step limitations We compute t by solving Eq. (64). (iv) Upate the geometrical quantities We compute (X,n+1, Y,n+1 ) from Eq. (60). Then, we euce L +,n+1 an N +,n+1. (v) Upate the physical variables We get τ,n+1, U,n+1, an E,n+1 from Eqs.(61) (63).

18 1346 uan-wen GE (vi) Equation of state The internal energy ε,n+1 is given by ε,n+1 = E,n+1 U,n+1 2 /2. Then, we get the pressure P,n+1 an the isentropic soun spee a,n+1 from the equation of state. 4 Numerical results 4.1 Saltzman s shock tube We consier the movement of a planar shock wave on the artesian gris [1]. This is a well-known ifficult test case that can valiate the robustness of the present scheme when the mesh is not aligne with the flui flow. The computational omain is the rectangle (x, y) [0, 1] [0, 0.1]. The initial mesh is obtaine by transforming a uniform cells with the mapping { Xstr = x + (0.1 y)sin(πx), Y str = y. For the material, we use the equation of state of the monoatomic gas (γ = 5/3). The initial state is (ρ 0, P 0, U0 )=(1, 0, 0). The bounary conition at x = 0 is a normal velocity U = 1 (the inflow velocity). On all the other bounaries, we set the wall conitions. Figure 9 shows the initial gris. The exact solution is a planar shock wave that moves at the spee D = 4/3 from left to right. The correct answer for the ensity is 4.0 in the singly shocke region, 10.0 in the oubly shocke region an 20.0 in the triply shocke region. The locations of the shock front are 0.967, 0.927, an 0.96 at t =0.80, t =0.86, an t =0.93, respectively. Figures 10 an 11 show the gris at t =0.80 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme, respectively. Figures 12 an 13 show the contour plots of the ensity at t = 0.80 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme. The gris at t= 0.86 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme are shown in Figs. 14 an 15. The contour plots of the ensity at t= 0.86 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme are shown in Figs.16 an 17. The gris at t= 0.93 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme are shown in Figs. 18 an 19. The contour plots of the ensity at t= 0.93 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme are shown in Figs. 20 an 21. All gri istortion is spurious since this is a purely one-imensional problem. The comparison shows that the Lagrangian cell-centere conservative scheme preserves the Fig. 9 Initial gris for Saltzman s shock tube Fig. 10 Gris for Maire et al. [1] scheme at t= 0.80 Fig. 11 Gris for Lagrangian cell-centere conservative scheme at t=0.80

19 Lagrangian cell-centere conservative scheme 1347 Fig. 12 ontour plots of ensity for Maire et al. [1] scheme at t=0.80 Fig. 13 ontour plots of ensity for Lagrangian cell-centere conservative scheme at t= 0.80 Fig. 14 Gris for Maire et al. [1] scheme at t= 0.86 Fig. 15 Gris for Lagrangian cell-centere conservative scheme at t=0.86 Fig. 16 ontour plots of ensity for Maire et al. [1] scheme at t=0.86 Fig. 17 ontour plots of ensity for Lagrangian cell-centere conservative scheme at t= 0.86

20 1348 uan-wen GE one-imensional solution very well (all layers in the vertical irection) an eliminates the artificial gri istortion an hourglass type motion. The answers are extremely close to the true solution. The robustness of the Lagrangian cell-centere conservative scheme is clearly emonstrate by this test case. The Maire et al. [1] scheme oes not preserve the one-imensional solution, an the shock face is increasing tilte. This error comes probably from the fact that the Maire et al. [1] scheme is not the momentum conservation scheme. Fig. 18 Gris for Maire et al. [1] scheme at t= 0.93 Fig. 19 Gris for Lagrangian cell-centere conservative scheme at t=0.93 Fig. 20 ontour plots of ensity for Maire et al. [1] scheme at t=0.93 Fig. 21 ontour plots of ensity for Lagrangian cell-centere conservative scheme at t= Shock refraction problem We now consier a shock refraction problem in the artesian geometry using the same gris as in Ref. [14]. In this problem, a piston moves from the left sening a shock wave through an initially col material of a unit ensity with a gri that is tapere so that its right bounary is at an angle of 60 relative to the vertical irection. This shock then becomes incient on a secon gri that is slante but uniformly space at an angle of 60. Reflective bounary conitions are applie to the top an bottom of both gris. The region compose of the secon gri has an initial ensity of 1.5. The problem runs for t= 1.30 just before the shock starts to run off the secon gri. In Fig.22, the initial gris are shown. Figures 23 an 24 epict the gris at t=1.30 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme.

21 Lagrangian cell-centere conservative scheme 1349 Figures 25 an 26 show the contour plots of the ensity at t= 1.30 for the Maire et al. [1] scheme an the Lagrangian cell-centere conservative scheme. The importance of this problem is that we have both the physical vorticity an the spurious gri istortion [4,6,14]. The gri in Fig. 23 is noticeably istorte, particularly in the vicinity of the lower part of the interface, where the gri is on the verge of tangling. Figure 24 has no spurious gri istortion an hourglass type motion. The contour plots of the ensity in Fig. 25 o not present the physical vorticity, while the contour plots of the ensity in Fig. 26 present the physical vorticity. These results o not permit one to istinguish between the two computations. However, the Lagrangian cell-centere conservative scheme is clearly superior from the point of view of robustness of the computational physical vorticity an the quality. Fig. 22 Initial gris for shock refraction problem Fig. 23 Gris for Maire et al. [1] scheme at t =1.30 Fig. 24 Gris for Lagrangian cell-centere conservative scheme at t =1.30 Fig. 25 ontour plots of ensity for Maire et al. [1] scheme at t=1.30 Fig. 26 ontour plots of ensity for Lagrangian cell-centere conservative scheme at t =1.30

22 1350 uan-wen GE 5 onclusions In this paper, a Lagrangian cell-centere conservative scheme is evelope. This scheme relies on a genuinely two-imensional noal solver. The main feature of the algorithm is the introuction of a piecewise constant pressure of cells. Using a piecewise constant pressure of cells, we construct the scheme, for which the momentum an the total energy are conservative. The vertex velocities an the numerical fluxes through the cell interfaces are not compute inepenently contrary to stanar approaches, but they are evaluate in a consistent manner ue to an original solver locate at the noes. The Lagrangian cell-centere conservative scheme has goo robustness accoring to the numerical results obtaine from the test cases presente in this paper. The future research is to evelop the higher-orer extension of the present scheme. References [1] Maire, P. H., Abgrall, R., Breil, J., an Ovaia, J. A cell-centere Lagrangian scheme for compressible flow problems. SIAM J. Sci. omput., 29(4), (2007) [2] Von Neumann, J. an Richtmyer, R. D. A metho for the numerical calculations of hyroynamics shocks. J. Appl. Phys., 21, (1950) [3] Wilkins, M. L. alculation of elastic plastic flow. Methos in omputational Physics (e. Aler, B.), Vol.3, Acaemic Press, New York (1964) [4] aramana, E. J. an Shashkov, M. J. Elimination of artificial gri istorsion an hourglass-type motions by means of Lagrangian subzonal masses an pressures. J. omput. Phys., 142, (1998) [5] aramana, E. J., Shashkov, M. J., an Whalen, P. P. Formulations of artificial viscosity for multiimensional shock wave computations. J. omput. Phys., 144, (1998) [6] ampbell, J.. an Shashov, J.. A tensor artificial viscosity using a mimetic finite ifference algorithm. J. omput. Phys., 172, (2001) [7] aramana, E. J., Burton, D. E., Shashov, M. J., an Whalen, P. P. The construction of compatible hyroynamics algorithms utilizing conservation of total energy. J. omput. Phys., 146, (1998) [8] ampbell, J.. an Shashov, M. J. A compatible Lagrangian hyroynamics algorithm for unstructure gris. Selcuk J. Appl. Math., 4(2), (2003) [9] Scovazzi, G., hriston, M. A., Hughes, T. J. R., an Shai, J. N. Stabilize shock hyroynamics: I. a Lagrangian metho. omput. Methos Appl. Mech. Engrg., 196, (2007) [10] Scovazzi, G. Stabilize shock hyroynamics: II. esign an physical interpretation of the SUPG operator for Lagrangian computations. omput. Methos Appl. Mech. Engrg., 196, (2007) [11] Scovazzi, G., Love, E., an Shashkov, M. J. Multi-scale Lagrangian shock hyroynamics on 1/P 0 finite elements: theoretical framework an two-imensional computations. omput. Methos Appl. Mech. Engrg., 197, (2008) [12] Gounov, S. K., Zabroine, A., Ivanov, M., Kraiko, A., an Prokopov, G. Résolution Numérique es Problèmes Multiimensionnels e la Dynamique es Gaz, Eitions Mir, Moscow (1979) [13] Aessio, F. L., arroll, D. E., Dukowicz, J. K., Johnson, J. N., Kashiwa, B. A., Maltru, M. E., an Ruppel, H. M. aveat: a omputer oe for Flui Dynamics Problems with Large Distortion an Internal Slip, Technical Report LA MS, Los Alamos National Laboratory (1986) [14] Dukowicz, J. K. an Meltz, B. Vorticity errors in multiimensional Lagrangian coes. J. omput. Phys., 99, (1992) [15] Despŕes, B. an Mazeran,. Lagrangian gas ynamics in two imensions an Lagrangian systems. Arch. Rational Mech. Anal., 178, (2005) [16] arré, G., Delpino, S., Despŕes, B., an Labourasse, E. A cell-centere Lagrangian hyroynamics scheme on general unstructure meshes in arbitrary imension. J. omput. Phys., 228, (2009)