One-dimensional Navier-Stokes finite element flow model

One-Dimensional Navier-Stokes Finite Element Flow Model Taha Sochi∗ Technical Report April 9, 2013 ∗ Imaging Sciences & Biomedical Engineering, King’s College London, St Thomas’ Hospital, London, SE1 7EH, UK. Email: taha.sochi@kcl.ac.uk. 1 Contents Contents 2 List of Tables 3 Abstract 4 1 Introduction 5 2 Theoretical Background 6 3 Weak Form of 1D Flow Equations 8 4 Finite Element Solution 9 4.1 Single Vessel Time-Independent Flow . . . . . . . . . . . . . . . . . 9 4.2 Single Vessel Time-Dependent Flow . . . . . . . . . . . . . . . . . . 14 4.3 Branched Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Non-Dimensionalized Form 18 5.1 Non-Dimensionalized Navier-Stokes Equations . . . . . . . . . . . . 18 5.2 Non-Dimensionalized Compatibility Condition . . . . . . . . . . . . 20 5.3 Non-Dimensionalized Matching Conditions . . . . . . . . . . . . . . 21 5.4 Non-Dimensionalized Boundary Conditions . . . . . . . . . . . . . . 21 6 Validation 21 7 General Notes 23 7.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.3 Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . 25 7.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 7.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 8 Conclusions 32 Nomenclature 33 References 35 9 Appendix A: Derivation of Analytical Solution 41 10 Appendix B: Gauss Quadrature 44 11 Appendix C: Biological Parameters 46 List of Tables 1 Gauss quadrature points and weights for polynomial order, p, 1-10 assuming a 0-1 master element. . . . . . . . . . . . . . . . . . . . . 3 45 4 Abstract This technical report documents the theoretical, computational, and practical aspects of the one-dimensional Navier-Stokes finite element flow model. The document is particularly useful to those who are interested in implementing, validating and utilizing this relatively-simple and widely-used model. Keywords: one-dimensional flow; Navier-Stokes; Newtonian fluid; finite element; elastic vessel; interconnected network; blood flow in large vessels; branching flow; time-independent flow; time-dependent flow. 1 INTRODUCTION 1 5 Introduction The one-dimensional (1D) Navier-Stokes flow model in its analytic formulation and numeric implementation is widely used for calculating and simulating the flow of Newtonian fluids in large vessels and in interconnected networks of such vessels [1– 5]. In particular, the model is commonly used by bioengineers to analyze blood flow in the arteries and veins. This model can be easily implemented using a numeric meshing technique, such as finite element method, to provide a computational framework for flow simulation in large tubes. The model can also be coupled with a pressure-area constitutive relation and hence be extended to elastic vessels and networks of elastic vessels. Despite its simplicity, the model is reliable within its validity domain and hence it can provide an attractive alternative to the more complex and costly multi-dimensional flow models in some cases of flow in regular geometries with obvious symmetry. The roots of the 1D flow model may be traced back to the days of Euler who apparently laid down its mathematical foundations. In the recent years, the 1D model became increasingly popular, especially in the hemodynamics modeling. This is manifested by the fact that several researchers [2–4, 6–18] have used this model recently in their modeling and simulation work. The ‘1D’ label attached to this model stems from the fact that the θ and r dependencies of a cylindrically-coordinated vessel are neglected due to the axisymmetric flow assumption and the simplified consideration of the flow profile within a lumped parameter called the momentum correction factor. Therefore, the only dependency that is explicitly accounted for is the dependency in the flow direction, x. The biggest advantages of the 1D model are the relative ease of implementation, and comparative low computational cost in execution. Therefore, the use of full multi-dimensional flow modeling, assuming its viability within the available 6 2 THEORETICAL BACKGROUND computational resources, is justified only when the 1D model fails to capture the essential physical picture of the flow system. However, there are several limitations and disadvantages of the 1D model that restrict its use. These limitations include, among other things, the Newtonian assumption, simplified flow geometry and the one-dimensional nature. 2 Theoretical Background The widely-used one-dimensional form of the Navier-Stokes equations to describe the flow in a vessel; assuming laminar, incompressible, axi-symmetric, Newtonian, fully-developed flow with negligible gravitational body forces; is given by the following continuity and momentum balance relations with suitable boundary conditions ∂A ∂Q + = 0 ∂t ∂x   ∂ αQ2 A ∂p Q ∂Q + + +κ = 0 ∂t ∂x A ρ ∂x A t ≥ 0, x ∈ [0, L] (1) t ≥ 0, x ∈ [0, L] (2) In these equations, A is the vessel cross sectional area, t is the time, Q is the volumetric flow rate, x is the axial coordinate along the vessel, L is the length of the vessel, α (= R u2 dA Au2 with u and u being the fluid local and mean axial speed respectively) is the momentum flux correction factor, ρ is the fluid mass density, p is the local pressure, and κ is a viscosity friction coefficient which is usually given by κ = 2παν/(α − 1) with ν being the fluid kinematic viscosity defined as the ratio of the dynamic viscosity µ to the mass density. These equations supported by appropriate compatibility and matching conditions are used to describe the 1D flow in a branched network of vessels. The equations, being two in three variables, Q A and p, are normally coupled with the following pressure-area relation in a distensible vessel to close the system and obtain a solution 7 2 THEORETICAL BACKGROUND p = po + f (A) (3) In this relation, p and po are the local and reference pressure respectively, and f (A) is a function of area which may be modeled by the following relation f (A) = p  β √ A − Ao Ao where β= (4) √ πho E 1 − ς2 (5) In these equations, Ao and ho are respectively the vessel cross sectional area and wall thickness at reference pressure po , while E and ς are the Young’s elastic modulus and Poisson’s ratio of the vessel wall. Similar variants of this 1D flow model formulation can also be found in the literature (see for example [7, 10, 18, 19]). The continuity and momentum equations are usually casted in matrix form [2, 11, 12, 16] which is more appropriate for numerical manipulation and discretization. In matrix form these equations are given by ∂U ∂F + +B=0 ∂t ∂x (6) where    A  U=  , Q and   F= αQ2 A + Q R a df da A ρ da     = Q αQ2 A + β A3/2 3ρAo    (7) 8 3 WEAK FORM OF 1D FLOW EQUATIONS    0  B=  κQ A (8) It should be remarked that the second term of the second row of the F matrix can be obtained from the third term of the original momentum equation as follow A ∂f A ∂f ∂A ∂ A ∂p = = = ρ ∂x ρ ∂x ρ ∂A ∂x ∂x ∂ = ∂x 3 Z A′ ∂ A ∂f ∂A = ρ ∂A ∂x Z A′ Z x′ A ∂f ∂A ∂x ρ ∂A ∂x ∂ A df dA = ρ dA ∂x  β A3/2 3ρAo  (9) Weak Form of 1D Flow Equations On multiplying Equation 6 by weight functions and integrating over the solution domain, x, the following is obtained Z Ω ∂U · ωdx + ∂t Z Ω ∂F · ωdx + ∂x Z Ω B · ωdx = 0 (10) where Ω is the solution domain, and ω is a vector of arbitrary test functions. On integrating the second term of Equation 10 by parts, the following weak form of the preceding 1D flow system is obtained Z Ω ∂U · ωdx − ∂t Z dω F· dx + dx Ω Z Ω B · ωdx + [F · ω]∂Ω = 0 (11) where ∂Ω is the boundary of the solution domain. This weak formulation, coupled with suitable boundary conditions, can be used as a basis for finite element implementation in conjunction with an iterative scheme such as Newton-Raphson method. 9 4 FINITE ELEMENT SOLUTION 4 Finite Element Solution There are two major cases to be considered in the finite element solution of the stated flow problem: single vessel and branched network where each one of these cases can be time-independent or time-dependent. These four cases are outlined in the following three subsections. 4.1 Single Vessel Time-Independent Flow The single vessel time-independent model is based on dropping the time term in the continuity and momentum governing equations to obtain a steady-state solution. This should be coupled with pertinent boundary and compatibility conditions at the vessel inlet and outlet. The details are given in the following. In discretized form, Equation 11 without the time term can be written for each node Ni (Ai , Qi ) as      fi   0  Ri =  =  0 gi (12) where R is a vector of the weak form of the residuals and fi = Nq  X q=1  ∂x dωAi dζ −wq Q(ζq ) (ζq ) (ζq ) (ζq ) + Q(∂Ω)ωAi (∂Ω) ∂ζ dζ dx (13) and gi Nq X     β dζ Q(ζq ) αQ2 (ζq ) dωQi ∂x 3/2 + A (ζq ) (ζq ) (ζq ) + κ ωQi (ζq ) = wq (ζq ) − ∂ζ A(ζ ) 3ρA dζ dx A(ζ ) q o q q=1   αQ2 (∂Ω) β 3/2 + + A (∂Ω) ωQi (∂Ω) (14) A(∂Ω) 3ρAo where q is an index for the Nq quadrature points, ζ is the quadrature point coor- 10 4.1 Single Vessel Time-Independent Flow dinate and A(ζq ) = n X Aci ψAi (ζq ) & Q(ζq ) = i n X Qci ψQi (ζq ) (15) i with n being the number of nodes in a standard element. Because of the nonlinear nature of the problem, an iteration scheme, such as Newton-Raphson, can be utilized to construct and solve this system of equations based on the residual. The essence of this process is to solve the following equation iteratively and update the solution until a convergence criterion based on reaching a predefined error tolerance is satisfied J ∆U = −R (16) In this equation, J is the jacobian matrix, ∆U is the perturbation vector, and R is the weak form of the residual vector. For a vessel with n nodes, the Jacobian matrix, which is of size 2n × 2n, is given by       J=      ∂f1 ∂A1 ∂f1 ∂Q1 ∂g1 ∂A1 ∂g1 ∂Q1 .. . .. . ∂fn ∂A1 ∂gn ∂A1 ··· ∂f1 ∂An ∂f1 ∂Qn ··· .. . ∂g1 ∂An ∂g1 ∂Qn .. . .. . ∂fn ∂Q1 ··· ∂fn ∂An ∂fn ∂Qn ∂gn ∂Q1 ··· ∂gn ∂An ∂gn ∂Qn             (17) where the subscripts stand for the node indices, while the vector of unknowns, which is of size 2n, is given by 11 4.1 Single Vessel Time-Independent Flow       U=      A1 Q1 .. . An Qn             (18) In the finite element implementation, the Jacobian matrix is usually evaluated numerically by finite differencing, i.e.       J≃      ∆f1 ∆A1 ∆f1 ∆Q1 ∆g1 ∆A1 ∆g1 ∆Q1 .. . .. . ∆fn ∆A1 ∆gn ∆A1 ··· ∆f1 ∆An ∆f1 ∆Qn ··· ... ∆g1 ∆An ∆g1 ∆Qn .. . .. . ∆fn ∆Q1 ··· ∆fn ∆An ∆fn ∆Qn ∂gn ∆Q1 ··· ∆gn ∆An ∆gn ∆Qn             (19) The procedure to obtain a solution is summarized in the following scheme 1. Start with initial values for Ai and Qi in the U vector. 2. The system given by Equation 16 is constructed where the weak form of the residual vector R may be calculated in each iteration l (= 0, 1, . . . , M ) as  f1 (Ul )       g1 (Ul )      .  .. Rl =         fn (Ul )    gn (Ul ) 3. The jacobian matrix is calculated from Equation 19. 4. System 16 is solved for ∆U, i.e. (20) 12 4.1 Single Vessel Time-Independent Flow ∆U = −J−1 R (21) 5. U is updated to obtain a new U for the next iteration, that is Ul+1 = Ul + ∆U (22) 6. The norm of the residual vector is calculated from p ǫ21 + ǫ22 + · · · + ǫ2N N= N (23) where ǫi is the ith entry of the residual vector and N (= 2n) is the size of the residual vector. 7. This cycle is repeated until the norm is less than a predefined error tolerance (e.g. 10−8 ) or a certain number of cycles is reached without convergence. In the last case, the operation will be aborted due to failure and may be resumed with improved finite element parameters. With regard to the boundary conditions (BC), two types of Dirichlet conditions can be applied: pressure and volumetric flow rate, that is A − ABC = 0 (for area BC) & Q − QBC = 0 (for flow BC) (24) These conditions are imposed by replacing the residual function of one of the governing equations (the continuity equation in our model) for the boundary nodes with one of these constraints. Imposing the boundary conditions as constraints in one of the two governing equations is associated with imposing compatibility conditions, arising from pro- 13 4.1 Single Vessel Time-Independent Flow jecting the differential equations in the direction of the outgoing characteristic variables [20], at the inlet and outlet by replacing the residual function contributed by the other governing equation with these conditions. The compatibility conditions are given by T l1,2  ∂U H +B ∂x  =0 (25) where H is the matrix of partial derivative of F with respect to U, while the transposed left eigenvectors of H are given by T l1,2  = −α Q A ± q Q2 A2 (α2 − α) + A df ρ dA  1 (26) that is H i.e.   0 1  ∂U  +B=  2 ∂x β Q 1/2 + A 2α −α Q A2 2ρAo A H  ∂U  +B=  2 ∂x + −α Q A2 −α Q A that is ± q Q2 A2 (α2 − α) + A df ρ dA 1  ∂Q ∂x   β A1/2 2ρAo      ∂A ∂x    0  +  κQ A ∂Q ∂x Hence, Equation 25 reduces to  ∂A ∂x + 2α ∂Q +κ ∂x (27)  (28) 
Q  A ∂Q ∂x 2 −α Q A2 + β A1/2 2ρAo  ∂A ∂x + 2α ∂Q ∂x +κ  
Q =0 A (29) 14 4.2 Single Vessel Time-Dependent Flow Q −α ± A s Q2 2 A df (α − α) + A2 ρ dA !     ∂A Q2 ∂Q ∂Q β Q 1/2 + −α 2 + A + 2α +κ =0 ∂x A 2ρAo ∂x ∂x A (30) In the last relation, the minus sign is used for the inflow boundary while the plus sign for the outflow boundary. The compatibility conditions, given by Equation 30, replace the momentum residual at the boundary nodes. 4.2 Single Vessel Time-Dependent Flow The aforementioned time-independent formulation can be extended to describe transient states by including the time terms in the residual equations in association with a numerical time-stepping method such as forward Euler, or backward Euler or central difference. The time-dependent residual will then be given (in one of the aforementioned schemes) by Rt+∆t TD = Z Ω Ut+∆t − Ut =0 · ωdx + Rt+∆t TI ∆t (31) where R is the weak form of the residual, T D stands for time-dependent and T I for time-independent. The time-dependent jacobian follows ∂Rt+∆t TD ∂Ut+∆t (32) ∆U = −J−1 R (33) Ul+1 = Ul + ∆U (34) Jt+∆t TD = Again, we have and 15 4.3 Branched Network where the symbols represent time-dependent quantities and l represents Newton iterations. With regard to the boundary nodes, a steady-state or time-dependent boundary conditions could be applied depending on the physical situation while a timedependent compatibility conditions should be employed by adding a time term to the time-independent compatibility condition, that is T CT D = l1,2 ∂U + CT I = 0 ∂t (35) where CT I is the time-independent compatibility condition as given by Equation 30, while CT D is the time-dependent compatibility condition, that is CT D =  ± q Q −α ± A s −α Q A Q2 A2 (α2 − α) + A df ρ dA CT D = ∂A ∂t  1 ! ∂A ∂Q + + CT I = 0 ∂t ∂t i.e. A df Q2 2 (α − α) + 2 A ρ dA     ∂Q ∂t   + CT I = 0 (36) (37) where the time derivatives can be evaluated by finite difference, e.g. At+∆t − At ∂A ≃ ∂t ∆t & ∂Q Qt+∆t − Qt ≃ ∂t ∆t (38) A sign convention similar to that outlined previously should be followed. An algorithmic code of the time-dependent module is presented in Algorithm 1. 4.3 Branched Network To extend the time-independent and time-dependent single vessel model to timeindependent and time-dependent branched network of interconnected vessels, matching constraints at the branching nodes are required. These nodes are treated as 16 4.3 Branched Network Initialize time: t = to Initialize Uto for j ← 1 to numberOf T imeSteps do Increment time by ∆t for i ← 1 to M aximumN umberOf N ewtonIterations do R Ut+∆t −U t = Find Rt+∆t =0 · ωdx + Rt+∆t TD TI ∆t Ω ∂Rt+∆t TD Find Jt+∆t T D = ∂Ut+∆t t+∆t Find ∆Ut+∆t = − (J−1 R)T D t+∆t Find Ut+∆t + ∆Ut+∆t i+1 = Ui t+∆t t+∆t Update: Ui = Ui+1 if (convergence condition met) then Exit loop else if (M aximumN umberOf N ewtonIterations reached) then Declare failure Exit end if end if end for Update: Ut = Ut+∆t end for Solution: Ut+∆t End Algorithm 1: Algorithmic code for the time-dependent module. discontinuous joints where each segment connected to that junction has its own index for that junction although they are spatially identical. The matching constraints are derived from the conservation of flow rate for incompressible fluid, and the Bernoulli energy conservation principle for inviscid flow. More specifically, at each n-segment branching node, n distinctive constraints are imposed: one represents the conservation of flow which involves all the segments at that junction, while the other (n − 1) constraints represent the Bernoulli principle with each Bernoulli constraint involving two distinctive segments. These constrains are summarized in 17 4.3 Branched Network the following relations n X Qi = 0 (39) i=1 and 1 1 pk + ρu2k − pl − ρu2l = 0 2 2 where k and l are indices of two distinct segments, and u (= (40) Q ) A is the fluid speed averaged over the vessel cross section. In Equation 39 a directional flow is assumed by attaching opposite signs to the inflow and outflow. The matching constraints, which replace the residuals of one of the governing equations (continuity), are coupled with compatibility conditions, similar to the ones used for the single vessel, where these conditions replace the residual of the other governing equation (momentum). The sign convention for these compatibility conditions should follow the same rules as for the boundary conditions, that is minus sign for inflow and plus sign for outflow. This branching model can be applied to any branching node with connectivity n ≥ 2. The special case of n = 2 enables flexible modeling of discontinuous transition between two neighboring segments with different cross sectional areas. Suitable pressure or flux boundary conditions (which for the timedependent case could be time-independent, or time-dependent over the whole or part of the time stepping process) should also be imposed on all boundary nodes of the network. With regard to the other aspects of the time-independence and time-dependence treatment, the network model should follow the same rules as for single vessel time-independent and time-dependent models which are outlined in the previous sections. 18 5 NON-DIMENSIONALIZED FORM 5 Non-Dimensionalized Form To improve convergence, the aforementioned dimensional forms of the governing, boundary, compatibility, and matching equations can be non-dimensionalized by carefully-chosen scale factors. The following scale factors are commonly used to scale the model parameters: Q ∼ πRo2 Uo A ∼ πRo2 p ∼ ρUo2 x∼λ t∼ λ Uo (41) where Ro , Uo , and λ are respectively typical values of the radius, velocity and length for the flow system. In the following we demonstrate non-dimensionalization of the flow equations by a few examples followed by stating the non-dimensionalized form of the others. 5.1 Non-Dimensionalized Navier-Stokes Equations Continuity equation 1st form (Equation 1): ∂A ∂Q + =0 ∂t ∂x that is ∂ (πRo2 A′ ) ∂ (πRo2 Uo Q′ )  +  =0 ′) λ ′ ∂ (λx ∂ Uo t ∂A′ ∂Q′ + =0 ∂t′ ∂x′ where the prime indicates a non-dimensionalized value. Continuity equation 2nd form (Equation 6): (42) (43) (44) 5.1 Non-Dimensionalized Navier-Stokes Equations 19 Same as Equation 44. Momentum equation 1st form (Equation 2): ∂Q ∂ + ∂t ∂x ∂ ∂ (πRo2 Uo Q′ )  +  ∂ (λx′ ) ∂ Uλo t′  αQ2 A α (πRo2 Uo Q′ ) (πRo2 A′ )  2 + ! + A ∂p Q +κ =0 ρ ∂x A (45) (πRo2 Uo Q′ ) (πRo2 A′ ) ∂ (ρUo2 p′ ) =0 + κ ρ ∂ (λx′ ) (πRo2 A′ ) (46) + πRo2 ρUo2 A′ ∂p′ κπRo2 Uo Q′ + =0 λρ ∂x′ πRo2 A′ (47)  + πRo2 Uo2 A′ ∂p′ κλπRo2 Uo2 Q′ + =0 λ ∂x′ λπRo2 Uo A′ (48)  + A′ ∂p′ κλ Q′ + =0 ∂x′ πRo2 Uo A′ (49) A′ ∂p′ 2πανλ Q′ + =0 ∂x′ (α − 1) πRo2 Uo A′ (50) A′ ∂p′ 2ανλ Q′ + =0 ∂x′ (α − 1) Ro2 Uo A′ (51) πRo2 Uo2 ∂Q′ απ 2 Ro4 Uo2 ∂ + λ ∂t′ πRo2 λ ∂x′  πRo2 Uo2 ∂Q′ απRo2 Uo2 ∂ + λ ∂t′ λ ∂x′  Q′2 A′ ∂ ∂Q′ +α ′ ′ ∂t ∂x  Q′2 A′ Q′2 A′ + ∂Q′ ∂ + α ∂t′ ∂x′  ∂Q′ ∂ +α ′ ′ ∂t ∂x  Q′2 A′   that is Q′2 A′  + Momentum equation 2nd form (Equation 6): 20 5.2 Non-Dimensionalized Compatibility Condition ∂ ∂Q + ∂t ∂x 2  αQ2 A  + β ∂ 3/2 Q A +κ =0 3ρAo ∂x A (52) 1/2 (πRo2 ) β ∂ ′3/2 πRo2 Uo Q′ + = 0 (53) A + κ λ3ρA′o ∂x′ πRo2 A′ πRo2 Uo2 ∂Q′ (πRo2 Uo ) ∂ + λ∂t′ λπRo2 ∂x′  πRo2 Uo2 ∂Q′ πRo2 Uo2 ∂ + λ∂t′ λ∂x′  Uo Q′ (πRo2 ) β ∂ ′3/2 A + κ =0 + λ3ρA′o ∂x′ A′ (54) ∂Q′ ∂ + ′ ′ ∂t ∂x β ∂ ′3/2 λ Q′ A + κ =0 πRo2 Uo A′ (πRo2 )1/2 Uo2 3ρA′o ∂x′ (55) 5.2  αQ′2 A′  + αQ′2 A′  αQ′2 A′  1/2 1 Non-Dimensionalized Compatibility Condition Time-independent term of compatibility condition: + −α Uo2 Q′2 A′2 s ! β Uo ∂Q′ Uo2 Q′2 2 A′ p √ (α − α) + ′2 ′ ′ 2 A λ∂x′ ρ πRo 2Ao A !   ∂A′ β πRo2 Uo ∂Q′ 2παν U o Q′ A′1/2 + p + 2α + ′ ′ 2 ′ λ∂x λ∂x α − 1 πRo2 A′ 2ρ πRo Ao −α Uo Q′ ± A′ = 0 (56) Time-dependent term of compatibility condition: Uo Q′ −α ± A′ s A′ Uo2 Q′2 2 β (α − α) + √ A′2 ρ (πRo2 )1/2 2A′o A′ ! Uo ∂Q′ ∂A′ + =0 ′ ∂t ∂t′ (57) 21 5.3 Non-Dimensionalized Matching Conditions 5.3 Non-Dimensionalized Matching Conditions Flow conservation: Q′1 − Q′2 − Q′3 = 0 (58) 1 2 1 2 p′k + u′ k − p′l − u′ l = 0 2 2 (59) Bernoulli: 5.4 Non-Dimensionalized Boundary Conditions A′ − A′BC = 0 6 (for area BC) & Q′ − Q′BC = 0 (for flow BC) (60) Validation The different modules of the the 1D finite element flow model are validated as follow • Time-independent single vessel: the numeric solution should match the analytic solution as given by Equation 63 which is derived in Appendix A. Also, the boundary conditions should be strictly satisfied. • Time-dependent single vessel: the solution should asymptotically converge to the analytic solution on imposing time-independent boundary conditions. Also, the boundary conditions should be strictly satisfied at all time steps. • Time-independent network: four tests are used to validate the numeric solution. First, the boundary conditions should be strictly satisfied. Second, the conservation of mass (or conservation of volume for incompressible flow), as given by Equation 39, should be satisfied at all branching nodes (bridge, 6 VALIDATION 22 bifurcation, trifurcation, etc.). A consequence of this condition is that the sum of the boundary inflow (sum of Q at inlet boundaries) should be equal to the sum of the boundary outflow (sum of Q at outlet boundaries). Third, the conservation of energy (Bernoulli’s principle for inviscid flow), as given by Equation 40, should be satisfied at all branching nodes. Fourth, the analytic solution for time-independent flow in a single vessel, as given by Equation 63 in Appendix A, should be satisfied by all vessels in the network with possible exception of very few vessels with odd features (e.g. those with distorted shape such as extreme radius-to-length ratio, and hence are susceptible to large numerical errors). The fourth test is based on the fact that the single vessel solution is dependent on the boundary conditions and not on the mechanism by which these conditions are imposed. • Time-dependent network: the solution is validated by asymptotic convergence to the time-independent solution, as validated by the four tests outlined in the previous item, on imposing time-independent boundary conditions. The solutions may also be tested qualitatively by static and dynamic visualization for time-independent and time-dependent cases respectively to verify their physical sensibility. Other qualitative tests, such as comparing the solutions of different cases with common features, may also be used for validation. It should be remarked that Equation 63 contains three variables: x A and Q, and hence it can be solved for one of these variables given the other two. Solving for A and Q requires employing a numeric solver, based for example on a bisection method; hence the best option is to solve for x and compare to the numeric solution. This in essence is an exchange of the role of independent and dependent variables which has no effect on validation. Alternatively, Equation 76 can be used to verify the solution directly by using the vessel inlet and outlet areas. In fact Equation 76 can be used to verify the solution at any point on the vessel axis by labeling the 23 7 GENERAL NOTES area at that point as Aou , as explained in Appendix A. 7 General Notes 7.1 Implementation • The model described in this report was implemented and tested on both single vessels and networks of vessels for time-independent and time-dependent cases and it produced valid results. The implementation is based on a Galerkin method, where the test functions are obtained from the same space as the trial basis functions used to represent the state variables, with a Lagrange interpolation associated with a Gauss quadrature integration scheme (refer to Appendix B for Gauss quadrature tables). Many tests have been carried out to verify various aspects of the 1D model. These tests involved many synthetic and biological networks which vary in their size, connectivity, number and type of branching nodes, type of meshing, and so on. The tests also included networks with and without loops although the great majority of the networks contain loops. Some of the networks involved in these tests consist of very large number of vessels in the order of hundreds of thousands with much more degrees of freedom. Non-dimensional form, as well as dimensional form, was tested on single vessels and branched networks; the results, after re-scaling, were verified to be identical to those obtained from the dimensional form. The checks also included h and p convergence tests which demonstrated correct convergence behavior. • To be on the safe side, the order of the quadrature should be based on the sum of orders of the interpolating functions, their derivatives and test functions. The adopted quadrature order scheme takes the highest order required by the terms. 24 7.2 Solution • A constant delta may be used in the evaluation of the Jacobian matrix by finite difference. A suitable value for delta may be ∆Ai = ∆Qi = 10−7 or 10−8 . 7.2 Solution • Negative flow in the solution means the flow direction is opposite to the vessel direction as indicated by the vessel topology, that is the flow rate of a segment indexed as N1 N2 will be positive if the flow is from N1 to N2 and negative if the flow is from N2 to N1 . • Interpolation polynomials of various degrees (p) in association with different meshing (h) should be used to validate the convergence behavior. The convergence to the correct solution should improve by increasing p and decreasing h. L2 error norm may be used as a measure for convergence; it is given by 2 L = Z 2 X (Sa − Sn ) dx 1/2 (61) where Sa and Sn are the analytic and numeric solutions respectively, and X is the solution domain. The integration can be performed numerically using, for example, trapezium or Simpson’s rules. The error norm should fall steadily as h decreases and p increases. • With regard to the previously outlined implementation of the 1D model (see § 7.1), typical solution time on a typical platform (normal laptop or desktop) for a single time-independent simulation on a typical 1D network consisting of hundreds of thousands of degrees of freedom is a few minutes. The final convergence is normally reached within 5-7 Newton iterations. The solution time of a single time step for the time-dependent case is normally less than 7.3 Non-Dimensionalization 25 the solution time of the equivalent time-independent case, and the number of the required Newton iterations of each time step in the time-dependent case is normally less than that for the corresponding time-independent case. • Since there are many sources of error and wrong convergence, each acquired solution should be verified by the aforementioned validation tests (see § 6). The 1D finite element code should be treated as a device that suggests solutions which can be accepted only if they meet the validation criteria. 7.3 Non-Dimensionalization • On implementing the non-dimensionalized form (as given in § 5) in the finite element code, all the user needs is to scale the primed input values either inside or outside the code; the results then should be scaled back to obtain the dimensionalized solution. • Different length scales can be utilized as long as they are in different orientations (e.g. vessel length and vessel radius) and hence linearly independent; otherwise the physical space will be distorted in non-systematic way and hence may not be possible to restore by scaling back. 7.4 Convergence A number of measures, outlined in the following points, can speedup convergence and help avoiding convergence failure. • Non-dimensionalization which requires implementation in the finite element code (as given in § 5) where the input data is non-dimensionalized and the results are re-dimensionalized back to the physical space. • Using different unit systems, such as m.kg.s or mm.g.s or m.g.s, for the input data and parameters. 7.5 Boundary Conditions 26 • Scaling the model up or down to obtain a similarity solution which can then be scaled back to obtain the final results. • Increasing the error tolerance of the solver for convergence criterion. However, the use of relatively large error tolerance can cause wrong convergence and hence should be avoided. It may be recommended that the maximum allowed error tolerance for obtaining a reliable solution must not exceed 10−5 . Anyway, the solution in all cases should be verified by the validation tests (see § 6) and hence it must be rejected if the errors exceed acceptable limits. • For time-dependent cases, the required boundary condition value can be imposed gradually by increasing the inlet pressure, for instance, over a number of time steps to reach the final steady state value. • The use of smaller time steps in the time-dependent cases may also help to avoid convergence failure. It should be remarked that the first three strategies are based on the same principle, that is adjusting the size of the problem numbers to help the solver to converge more easily to the solution. 7.5 Boundary Conditions • Dirichlet type boundary conditions are usually used for imposing flow rate and pressure boundary conditions. The previous formulation is based on this assumption. • Pressure boundary conditions are imposed by adjusting the inlet or outlet area where p and A are correlated through Equation 3. • While pressure boundary conditions can be imposed on both inlet and outlet boundaries simultaneously, as well as mixed boundary conditions (i.e. inlet 7.5 Boundary Conditions 27 pressure with outlet flow or inlet flow with outlet pressure or mixed on one or both boundaries), it is not possible to impose flow boundary conditions on all inlet and outlet boundaries simultaneously because this is either a trivial condition repeating the condition of flow conservation (i.e. Equation 39) at the branching junctions if the inflow is equal to the outflow or it is a contradiction to the flow conservation condition if the inflow and outflow are different, and hence no solution can be found due to ambiguity and lack of constraints in the first case and to inconsistency in the second case. • Zero Q boundary condition can be used to block certain inlet or outlet vessels in a network for the purpose of emulating a physical situation or improving convergence when the blockage does not affect the solution significantly. • In some biological flow conditions there are no sufficient data to impose realistic pressure boundary conditions that ensure biologically sensible flow in the correct direction over the whole vascular network. In such situations a back flow may occur in some branches which is physically correct but biologically incorrect. To avoid this situation, an inlet pressure boundary condition with outlet flow boundary conditions where the total outflow is split according to a certain physical or biological model (such as being proportional to the area squared) can be used to ensure sensible flow in the right direction over the whole network. The total amount of the outflow can be estimated from the inlet flow which is usually easier to estimate as it normally comes from a single (or few) large vessel. This trick may also be applicable in some physical circumstances. • Use may be made of an artificial single inlet boundary to avoid lack of knowledge about the pressure distribution in a multi-inlet network to ensure correct flow in the right direction. The inlets can be connected to a single artificial 28 7.6 Initial Conditions node (e.g. located at their centroid) where the radii of the connecting artificial vessels is chosen according to a physical or biological model such as Murray’s law. This node can then be connected through a single artificial vessel whose radius can be computed from a physical or biological model and whose length can be determined from a typical L/R ratio such as 10. The inlet of this vessel can then be used to impose a single p or Q biologicallysensible boundary condition. It should be remarked that Murray’s biological law is given by γ rm = n X rdγi (62) i where rm is the radius of the mother vessel, rdi is the radius of the ith daughter vessel, n is the number of daughter vessels which in most cases is 2, and γ is a constant index which according to Murray is 3, but other values like 2.1 and 2.2 are also used in the literature. • Time-dependent boundary conditions can be modeled by empirical signals (e.g. obtained from experimental data) or by closed analytical forms such as sinusoidal. 7.6 Initial Conditions • The convergence usually depends on the initial values of area and flow rate. A good option for these values is to use unstressed area with zero flow for start. 7.7 Miscellaneous • Apart from the interpolation nodes, there are two main types of nodes in the finite element network: segment nodes and finite element discretization 7.7 Miscellaneous 29 nodes. The connectivity of the second type is always 2 as these nodes connect two elements; whereas the connectivity of the first can be 1 for the boundary nodes, 2 for the bridge nodes connecting two segments, or ≥ 3 for the branching nodes (bifurcation, trifurcation, etc.). The mass and energy conservation conditions can be extended to include all the segment nodes with connectivity > 1 by including the bridge nodes. • For networks, the vessel wall thickness at reference pressure, ho , can be a constant or vary from vessel to vessel depending on the physical or biological situation. Using variable thickness is more sound in biological context where the thickness can be estimated as a fraction of the lumen or vessel radius. A fractional thickness of 10-15% of the radius is commonly used for blood vessels [6, 16, 20–26]. For more details, refer to Appendix D. • The previous finite element formulation of the 1D model for single vessels and networks works for constant-radius vessels (i.e. with constant Ao ) only and hence to extend the formulation to variable-radius vessels the previous matrix structure should be reshaped to include the effect of tapering or expanding of the vessels. However, the vessels can be straight or curved. The size of the vessels in a network can also vary significantly from one vessel to another as long as the 1D flow model assumptions (e.g. size, shape, etc.) do apply on each vessel. • The networks used in the 1D flow model should be totally connected, that is any node in the network can be reached from any other node by moving entirely inside the network vessels. • Different time stepping schemes, such as forward or backward Euler or central difference, can be used for implementing the time term of the time-dependent single vessel and network modules although the speed of convergence and 7.7 Miscellaneous 30 quality of solution vary between these schemes. The size of the time step should be chosen properly for each scheme to obtain equivalent results from these different schemes. • Although the 1D model works on highly non-homogeneous networks in terms of vessels length without discretization, a scheme of homogeneous discretization may be employed by using a constant element length, h, over the whole network as an approximation to the length of the discretized elements. The length of the elements of each vessel is then obtained by dividing the vessel evenly to an integer number of elements with closest size to the given h. Although discretization is not a requirement, since the 1D model works even on non-discretized networks, it usually improves the solution. Moreover, discretization is required for obtaining a detailed picture of the pressure and flow fields at the interior points. Use of interpolation schemes higher than linear (with and without discretization) also helps in refining the variable fields. Also, for single vessel the solution can be obtained with and without discretization; in the first case the discretized elements could be of equal or varying length. The solution, however, should generally improve by discretization. • Although the 1D model works on non-homogeneous networks in terms of vessels radius, an abrupt transition from one vessel to its neighbor may hinder convergence. • In general, the time-dependent problem converges more easily than its equivalent time-independent problem. This may be exploited to obtain approximate time-independent solutions in some circumstances from the time-dependent module as the latter asymptotically approaches the time-independent solution. 7.7 Miscellaneous 31 • The correctness of the solutions mathematically may not guarantee physiological, and even physical, sensibility since the network features, boundary conditions, and model parameters which in general highly affect the flow pattern, may not be found normally in real biological and physical systems. The quality of any solution, assuming its correctness in mathematical terms, depends on the quality of the underlying model and how it reflects the physical reality. • Because the 1D model depends on the length of the vessels but not their location or orientation, a 1D coordinate system, as well as 2D or 3D, can be used for coordinating the space. The vessels can be randomly oriented without effecting the solution. A multi-dimensional space may be required, however, for consistent and physically-correct description of the networks. • The reference pressure, po , in Equation 3 is usually assumed zero to simplify the relation. 8 CONCLUSIONS 8 32 Conclusions The one-dimensional Navier-Stokes formulation is widely used as a realistic model for the flow of Newtonian fluids in large vessels with certain simplifying assumptions, such as axi-symmetry. The model may also be coupled with a pressure-area constitutive relation and hence be extended to the flow in distensible vessels. Numerical implementation of this model based on a finite element method with suitable boundary conditions is also used to solve the time-independent and transient flow in single vessels and networks of interconnected vessels where in the second case compatibility and matching conditions, which include conservation of mass and energy, at branching nodes are imposed. Despite its comparative simplicity, the 1D flow model can provide reliable solutions, with relatively low computational cost, to many flow problems within its domain of validity. The current document outlined the analytical and numerical aspects of this model with theoretical and technical details related to implementation, performance, methods of improvement, validation, and so on. 8 CONCLUSIONS Nomenclature α correction factor for axial momentum flux β parameter in the pressure-area relation γ Murray’s law index ǫ residual error ζ quadrature point coordinate κ viscosity friction coefficient µ fluid dynamic viscosity ν fluid kinematic viscosity (ν = µρ ) ρ fluid mass density ς Poisson’s ratio of vessel wall ψ basis function for finite element discretization ω vector of test functions in the weak form of finite element Ω solution domain ∂Ω boundary of the solution domain A vessel cross sectional area ABC boundary condition for vessel cross sectional area Ain vessel cross sectional area at inlet Ao vessel cross sectional area at reference pressure B matrix of force terms in the 1D Navier-Stokes equations E Young’s modulus of vessel wall f (A) function in pressure-area relation F matrix of flux quantities in the 1D Navier-Stokes equations 33 8 CONCLUSIONS h length of element ho vessel wall thickness at reference pressure H matrix of partial derivative of F with respect to U J Jacobian matrix L length of vessel N norm of residual vector p local pressure p order of interpolating polynomial po reference pressure q dummy index for quadrature point Q volumetric flow rate QBC boundary condition for volumetric flow rate r radius R weak form of residual vector Sa analytic solution Sn numeric solution t time ∆t time step u local axial speed of fluid u mean axial speed of fluid U vector of finite element variables ∆U vector of change in U x vessel axial coordinate 34 REFERENCES 35 References [1] M.S. 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Computational Mechanics, 46(1):91–101, 2010. 46 41 9 APPENDIX A: DERIVATION OF ANALYTICAL SOLUTION 9 Appendix A: Derivation of Time-Independent Analytical Solution for Single Vessel The following analytical relation linking vessel axial coordinate x to cross sectional area A, cross sectional area at inlet Ain , and volumetric flow rate Q for timeindependent flow can be derived and used to verify the finite element solution αQ2 ln (A/Ain ) − x= β 5ρAo κQ  5/2 A5/2 − Ain  (63) The derivation is outlined in the following. For time-independent flow, the system given by Equation 6 in matrix form, will become ∂Q =0 ∂x ∂ ∂x  β αQ2 + A3/2 A 3ρAo x ∈ [0, l] , t ≥ 0  +κ Q =0 A x ∈ [0, l] , t ≥ 0 (64) (65) that is Q as a function of x is constant and ∂ ∂A  β αQ2 + A3/2 A 3ρAo  ∂A Q +κ =0 ∂x A (66) i.e.  β αQ2 − 2 + A1/2 A 2ρAo  ∂A Q +κ =0 ∂x A (67) which by algebraic manipulation can be transformed to 2 β + 2ρA A3/2 − αQ ∂x A o = ∂A −κQ On integrating the last equation we obtain (68) 42 9 APPENDIX A: DERIVATION OF ANALYTICAL SOLUTION x= β A5/2 5ρAo αQ2 ln A − κQ +C (69) where C is the constant of integration which can be determined from the boundary condition at x = 0 with A = Ain , that is C= −αQ2 ln Ain + 5/2 β A 5ρAo in (70) κQ i.e. αQ2 ln (A/Ain ) − x= β 5ρAo κQ  5/2 A5/2 − Ain  (71) For practical reasons, this relation can be re-shaped and simplify to reduce the number of variables by the use of the second boundary condition at the outlet, as outlined in the following. When x = L, A = Aou where L is the vessel length and Aou is the cross sectional area at the outlet, that is 2 L= αQ ln (Aou /Ain ) − β 5ρAo κQ  5/2 Aou − 5/2 Ain  (72) which is a quadratic polynomial in Q i.e. − α ln (Aou /Ain ) Q2 + κLQ + α ln (Ain /Aou ) Q2 + κLQ +  β  5/2 5/2 Aou − Ain = 0 5ρAo  β  5/2 5/2 Aou − Ain = 0 5ρAo (73) (74) with a solution given by Q= −κL ± r κ 2 L2 − 4α ln (Ain /Aou ) β 5ρAo 2α ln (Ain /Aou )  5/2 Aou − 5/2 Ain  (75) 9 APPENDIX A: DERIVATION OF ANALYTICAL SOLUTION 43 which is necessarily real for Ain ≥ Aou which can always be satisfied for normal flow conditions by proper labeling. For a flow physically-consistent in direction with the pressure gradient, the root with the plus sign should be chosen, i.e. Q= −κL + r   5/2 5/2 β A − A κ2 L2 − 4α ln (Ain /Aou ) 5ρA ou in o 2α ln (Ain /Aou ) (76) This, in essence, is a relation between flow rate and pressure drop (similar to the Hagen-Poiseuille law for rigid vessels) although for elastic vessels the flow rate, as given by Equation 76, does not depend on the pressure difference (as for rigid vessels) but on the actual inlet and outlet pressures. Although Equation 76 may look a special case of Equation 63 as it involves only the vessel two end areas, Aou may be assumed to be the area at any point along the vessel axis, with L being the distance form the vessel inlet to that point, and hence this relation can be used to verify the finite element solution at any point on the vessel. 10 APPENDIX B: GAUSS QUADRATURE 10 44 Appendix B: Gauss Quadrature In this appendix we list points and weights of Gauss quadrature for polynomials of order 1-10 which may not be easy to find. 0.500000000000000 0.211324865405188 0.112701665379259 0.069431844202974 0.046910077030669 0.033765242898424 0.025446043828621 0.019855071751232 0.015919880246187 0.013046735741414 0.788675134594812 0.500000000000000 0.330009478207572 0.230765344947159 0.169395306766867 0.129234407200303 0.101666761293187 0.081984446336682 0.067468316655508 0.887298334620741 0.669990521792428 0.500000000000000 0.380690406958402 0.297077424311301 0.237233795041836 0.193314283649705 0.160295215850488 0.930568155797026 0.769234655052841 0.619309593041598 0.500000000000000 0.408282678752175 0.337873288298095 0.283302302935377 1 2 3 4 5 6 7 8 9 10 1.000000000000000 0.500000000000000 0.277777777777778 0.173927422568727 0.118463442528095 0.085662246189585 0.064742483084435 0.050614268145188 0.040637194180787 0.033335672154344 0.500000000000000 0.444444444444444 0.326072577431273 0.239314335249683 0.180380786524069 0.139852695744638 0.111190517226687 0.090324080347429 0.074725674575291 0.277777777777778 0.326072577431273 0.284444444444444 0.233956967286345 0.190915025252559 0.156853322938943 0.130305348201467 0.109543181257991 0.173927422568727 0.239314335249683 0.233956967286345 0.208979591836735 0.181341891689181 0.156173538520001 0.134633359654998 Points 0.953089922969331 0.830604693233133 0.966234757101576 0.702922575688699 0.870765592799697 0.591717321247825 0.762766204958164 0.500000000000000 0.662126711701905 0.425562830509185 0.574437169490815 Weights 0.118463442528095 0.180380786524069 0.190915025252559 0.181341891689181 0.165119677500630 0.147762112357376 0.085662246189585 0.139852695744638 0.156853322938943 0.156173538520001 0.147762112357376 0.974553956171380 0.898333238706813 0.806685716350295 0.716697697064623 0.980144928248768 0.918015553663318 0.839704784149512 0.984080119753813 0.932531683344493 0.986953264258586 0.064742483084435 0.111190517226687 0.130305348201467 0.134633359654998 0.050614268145188 0.090324080347429 0.109543181257991 0.040637194180787 0.074725674575291 0.033335672154344 10 APPENDIX B: GAUSS QUADRATURE Table 1: Gauss quadrature points and weights for polynomial order, p, 1-10 assuming a 0-1 master element. p 1 2 3 4 5 6 7 8 9 10 45 11 APPENDIX C: BIOLOGICAL PARAMETERS 11 46 Appendix C: Biological Parameters In this appendix, we suggest some biologically-realistic values for the 1D flow model parameters in the context of simulating blood flow in large vessels. 1. Blood mass density (ρ): 1050 kg.m−3 [6, 7, 16, 25, 27–31]. 2. Blood dynamic viscosity (µ): 0.0035 Pa.s [4, 6, 7, 15, 16, 20, 25, 27, 30, 32–34]. 3. Young’s elastic modulus (E): 100 kPa [4, 6, 13, 15, 16, 20, 22, 25, 27, 34–38]. Also see [25, 39] on shear modulus. 4. Vessel wall thickness (ho ): this, preferably, is vessel dependent, i.e. a fraction of the lumen or vessel radius according to some experimentally-established mathematical relation. The relation between wall thickness and vessel inner radius is somehow complex and vary depending on the type of vessel (e.g. artery or capillary). For arteries, the typical ratio of wall thickness to inner radius is about 0.1-0.15, and this ratio seems to go down in the capillaries and arterioles. Therefore a typical value of 0.1 seems reasonable [6, 16, 20–26]. 5. Momentum correction factor (α): 4/3 = 1.33 assuming Newtonian flow. A smaller value, e.g. 1.2, may be used to account for non-Newtonian shearthinning effects [2, 3, 7, 16, 17, 20, 28, 40]. 6. Time step (∆t): 1.0-0.1 ms [4, 7, 9, 10, 13, 15, 16, 18–20, 25, 29, 34, 41–45]. 7. Pressure step (∆p): 1.0-5.0 kPa [4, 7, 16, 29]. 8. Time of heart beat: 0.85 s assuming 70 beats per minute. 9. Poisson ratio (ς): 0.45 [2, 4, 6, 13, 22, 25, 27, 34, 37, 39, 41].