Inexact Newton-type methods for non-linear problems arising from the SUPG/PSPG solution of steady incompressible navier-stokes equations

TECHNICAL PAPERS

Inexact Newton-type methods for non-linear problems arising from the SUPG/PSPG solution of steady incompressible navier-stokes equations

R. N. Elias; A. L. G. A. Coutinho; M. A. D. Martins

Center for Parallel Computations and Department of Civil Engineering, Federal University of Rio de Janeiro, P. O. Box 68506, 21945-970 Rio de Janeiro, Rj, Rio de Janeiro, RJ. Brazil, {renato, alvaro, marcos}@nacad.ufj.br

ABSTRACT

The finite element discretization of the incompressible steady-state Navier-Stokes equations yields a non-linear problem, due to the convective terms in the momentum equations. Several methods may be used to solve this non-linear problem. In this work we study Inexact Newton-type methods, associated with the SUPG/PSPG stabilized finite element formulation. The resulting systems of equations are solved iteratively by a preconditioned Krylov-space method such as GMRES. Numerical experiments are shown to validate our approach. Performance of the nonlinear strategies is accessed by numerical tests. We concluded that Inexact Newton-type methods are more efficient than conventional Newton-type methods.

Keywords: Inexact Newton-type methods, Newton-Krylov methods, Navier-Stokes, incompressible fluid flow, finite elements

Introduction

We consider the simulation of incompressible fluid flow governed by Navier-Stokes equations using the stabilized finite element formulation proposed by Tezduyar (1991). This formulation allows that equal-order-interpolation velocity-pressure elements are employed, circumventing the Babuska-Brezzi stability condition by introducing two stabilization terms. The first term is the Streamline Upwind Petrov-Galerkin (SUPG) presented by Brooks & Hughes (1982) and the other one is the Pressure Stabilizing Petrov Galerkin (PSPG) stabilization proposed initially by Hughes et al (1986) for Stokes flows and later generalized by Tezduyar (1991) to finite Reynolds number flows. SUPG/PSPG implementations have been very successful in the simulation of complex, large scale fluid flow problems in parallel computers (Tezduyar et al, 1993, 1996). Nowadays it has been used for simulations of 3D fluid-structure interaction problems on unstructured grids with more than one billion of elements (Aliabadi et al, 2002). Usually the resulting fully coupled (velocity-pressure) linearized systems of equations are solved by a preconditioned Krylov-space iterative method such as GMRES (Saad, 2003).

If we restrict ourselves to the steady case, it is known that, when discretized, the incompressible Navier-Stokes equations give rise to a system of nonlinear algebraic equations due the presence of convective terms in the momentum equations. Among several strategies to solve nonlinear problems the Newton's methods are attractive because it converges rapidly from any sufficient good initial guess (Dembo et al, 1982), (Kelley, 1995). However, the implementation of Newton's method involves some considerations: determining steps of Newton's method requires the solution of linear systems at each stage and exact solutions can be too expensive if the number of unknowns is large. In addition, the computational effort spent to find exact solutions for the linearized systems may not be justified when the nonlinear iterates are far from the solution. Therefore, it seems reasonable to use an iterative method (Barret et al, 1994), such as BiCGSTAB or GMRES, to solve these linear systems only approximately.

The inexact Newton method associated with a proper iterative Krylov solver, presents an appropriated framework to solve nonlinear systems, offering a trade-off between accuracy and amount of computational effort spent per iteration. For a proper mathematical description of the inexact Newton method we refer to Kelley (1995). It is often necessary to increase the robustness of the inexact Newton method adding some globalization procedure (Kelley, 1995, Pernice et al, 1998).

In the context of the SUPG/PSPG formulation for the steady incompressible Navier-Stokes equations coupled with heat and mass transfer, the work of Shadid et al (1997) investigated in depth the behavior of the inexact Newton method with backtracking. They have shown its computational efficiency and robustness. However, they pointed out that globalization by backtracking may be less robust when high accuracy is required at each linear solve than when less accuracy is required. A mathematical explanation for this behavior is provided in Tuminaro et al (2002). In the works of Shadid et al (1997) and Tuminaro et al (2002) the Jacobian was evaluated by a mixed method. Contributions emanating from the Galerkin term were computed analytically, whereas contributions from the stabilization terms were evaluated by numerical differentiation.

In this work, we evaluate the effectiveness of some Newton-type methods dealing with problems involving steady incompressible fluid flows. We also investigate the influence of the Jacobian form described by Tezduyar (1999). This form is based on Taylor's expansions of the nonlinear terms and presents an alternative and simple way to implement the tangent matrix employed by inexact Newton-type methods. The Krylov-subspace iterative driver of the Newton-type algorithms is a nodal-block diagonal preconditioned element-by-element GMRES solver. The test problems are the well-known driven cavity flow and flow over a backward facing step.

The remainder of this paper is organized as follows. The governing equations and the SUPG/PSPG finite element formulation are stated in Section 2. In Section 3 we present the inexact Newton and inexact successive substitution methods, showing how to choose the parameter used to force the residual to be suitably small, the backtracking strategy and the evaluation of the approximate Jacobian. In Section 4 parametric studies are presented for two test problems, accessing performance and accuracy of the nonlinear solution methods. The paper ends with a summary of our main conclusions.

Governing Equations and Finite Element Formulation

Let WÌ be the spatial domain, where n_sd is the number of space dimensions. Let G = G_gÈ G_hdenote the boundary of W. We consider the following velocity-pressure formulation of the Navier-Stokes equations governing steady incompressible flows,

where r and u are the density and velocity respectively, and s is the stress tensor given as

with,

Here, p and m are the pressure and dynamic viscosity, and I is the identity tensor. The essential and natural boundary conditions associated with equations (1) and (2) are represented as

Let us assume following Tezduyar (1991) that we have some suitably defined finite-dimensional trial solution and test function spaces for velocity and pressure, , , and = . The stabilized finite element formulation of equations (1) and (2) can then be written as follows: Find u^h Î and p^h Î such that "w^h Î and "q^h Î :

In the above equation the first three integrals on the left hand side and the right hand side integral represent terms that appear in the standard Galerkin formulation of the problem (1)-(6), while the remaining integral expressions represent the additional terms which arise from the stabilized finite element formulation of the problem. Note that the stabilization terms are evaluated as the sums of element-wise integral expressions. The first summation corresponds to the SUPG (Streamline Upwind Petrov/Galerkin) term and the second correspond to the PSPG (Pressure Stabilization Petrov/Galerkin) term. We have calculated the stabilization parameters (Tezduyar et al, 1992) as follows,

Here u^h is the local velocity vector, n represent the kinematic viscosity and h^# is the element length, defined to be equal to the diameter of the circle which is area-equivalent to the element. The spatial discretization of equation (7) leads to the following set of non-linear algebraic equations,

where u is the vector of unknown nodal values of u^h and p is the vector of unknown nodal values of p^h. The non-linear vectors N(u), N_g(u), and N_j(u) the matrices K, G, G_d and G_j emanate, respectively, from the convective, viscous and pressure terms. The vectors f_u and f_p are due to the boundary conditions (5) and (6). The subscripts g and j identify the SUPG and PSPG contributions respectively. In order to simplify the notation we denote by x = (u, p) a vector of nodal variables comprising both nodal velocities and pressures, thus, equation (9) can be written as,

where F(x) represents a nonlinear vector function.

For Reynolds numbers much greater than unity the nonlinear character of the equations becomes dominant, making the choice of the solution algorithm, especially with respect to its convergence and efficiency, a key issue. The search for a suitable nonlinear solution method is complicated by the existence of several procedures and their variants. In the following section we present the nonlinear solution strategies based on the Newton-type methods which are evaluated in this work.

Nonlinear Solution Procedures

Consider the nonlinear problem arising from the discretization of the fluid flow equations described by equation (10). We assume that F is continuously differentiable in and denote its Jacobian matrix by F' Î . The Newton's method is a classical algorithm for solving equation (10) and can be enunciated as: Given an initial guess x₀, we compute a sequence of steps s_k and iterates x_k as follows,

Newton's method is attractive because it converges rapidly from any sufficiently good initial guess (see Dembo et al, 1982). However, one drawback of Newton's method is the need to solve the Newton equations (11) at each stage. Computing the exact solution using a direct method can be expensive if the number of unknowns is large and may not be justified when x_k is far from a solution. Thus, one might prefer to compute some approximate solution, leading to the following algorithm:

for some h_k Î[0,1) where is a norm of choice. This formulation naturally allows the use of an iterative solver: one first chooses h_k and then applies the iterative solver to equation (11) until a s_k is determined for which the residual norm satisfies equation (12) In this context h_k is often called the forcing term, since its role is to force the residual of (11) to be suitably small. This term can be specified in several ways (see, Eisenstat & Walker, 1996, Shadid et al, 1997, Pernice et al, 1998) to enhance efficiency and convergence and will be treated in the following section below. In our implementation we have used the nodal block-diagonal preconditioned GMRES(m) method to solve the Newton equations (11), where m represents the number of basis vectors used by GMRES algorithm (Saad, 2003).

A particularly simple scheme for solving the nonlinear system of equations (10) is a fixed point iteration procedure also known as the successive substitution, the Picard iteration, functional iteration or successive iteration. Note in the algorithms above that if we do not build the Jacobian matrix in equations (11) and (12) and the solution of previous iterations were reused, we have a successive substitution (SS) method. In this work, we have evaluated the efficiency of Newton and successive substitution methods and their inexact versions. We may also define a mixed strategy combining SS and N (or ISS and IN) iterations, to improve performance, as discussed in the following. In this strategy the Jacobian evaluation is enabled after k successive substitutions. Thus, we have labeled the mixed strategy as k-SS+N or as k-ISS+IN in the case of its inexact counterpart.

Forcing Term

We have implemented the forcing term as a variation of the choice from Eisenstat & Walker (1996) that tends to minimize oversolving while giving fast asymptotic convergence to a solution of (10). Oversolving means that the linear equation for the Newton step is solved to a precision far beyond what is needed to correct the nonlinear iteration. Kelley (1995) considered the following measure of the degree to which the nonlinear iteration approximates the solution,

where g Î [0,1) is a parameter. In order to specify the choice at k = 0 and bound the sequence away from 1 we set

Here the parameter h_max is an upper limit of the sequence {}. We have chosen g = 0.9 according to Eisenstat & Walker (1996) and adopted h_max = 0.9, 0.5, 0.1 arbitrarily in our tests.

It may happen that is small for one or more iterations while x_k is still far from the solution. A method of safeguarding against this possibility was suggested by Eisenstat & Walker (1996) to avoid volatile decreases in h_k. The idea is that if h_k-1 is sufficiently large we do not let h_k decrease by much more than a factor of h_k-1, that is:

The constant 0.1 is arbitrary. According to Kelley (1995) the safeguarding does improve the performance of the iteration.

There is a chance that the final iterate will reduce far beyond the desired level, consequently the cost of the solution of the linear equation for the last step will be higher than really needed. This oversolving in the final step can be controlled comparing the norm of the current nonlinear residual to the nonlinear norm at which the iteration would terminate

and bounding h_k by a constant multiple of . We have used the choice proposed by Kelley (1995)

where t_NL represent the nonlinear tolerance.

Backtracking Procedures

The backtracking procedures are employed, in the context of nonlinear methods, to improve the convergence capability for any initial guess (Eisenstat & Walker, 1994). These procedures, also known as line searches or globalization procedures are based on trying to recover the convergence of a nonlinear method performing reductions on a Newton step according to the evolution of nonlinear residual. There are several forms to impose these step reductions and a constant reduction is often employed successfully, but sometimes disturbing the nonlinear convergence. In this work we employ the Armijo rule as described in Kelley (1995). This rule tries to compute a sufficient Newton step reduction without disturbing the nonlinear convergence, according to a residual decreasing function (see Kelley, 1995). Thus we may describe the Armijo rule by the procedures below.

After rejecting j step reductions we have the following sequence:

where l is the reduction factor that we are looking for. Note that the sequence above can be modeled by the scalar function,

The minimal of the function (19) can be used for computing the next step reduction. According to Kelley (1995), we have used a third order polynomial to compute the step reduction. This procedure can be summarized as: After reject l_c and build the polynomial model we compute the minimal of l_t analytically and the following adjustment is performed,

In a 3-point parabolic model, the procedure for evaluating f (0) and f (1) is performed by the following steps. If the step for l = 1 was rejected, adjust l = s₁ and try again. After the second step rejection, we have the following values to construct the 3-point parabolic model,

where l_c and l_j are the values of l most recently rejected. The interpolation polynomial of f at the points 0, l_c, l_j is,

We must evaluate two possibilities for the curvature of polynomial in equation (22). Thus we have,

and

If the curvature of p is positive, set l_t to the minimal of p and compute l_j+1 by equation (20); otherwise p'' (0) < 0, set l_j+1 as the minimal of p in the interval [s₀l, s₁l] or reject the parabolic model and set l _j+1 as s₀l or s₁l. In this work we adopted the second strategy, setting l_j+1 = s₁l.

Jacobian Matrix Evaluation

To form the Jacobian F' required by Newton-type methods we use a numerical approximation described by Tezduyar (1999). Consider the following Taylor expansion for the nonlinear convective term emanating from the Galerkin formulation:

where Du is the velocity increment. Discarding the high order terms and omitting the integral symbols we arrive to the following approximation,

Note that the first term in the right hand side of equation (26) is the corresponding residual vector and the remaining terms represent the numerical approximation of . If we apply similar derivations to N_g(u), and N_j(u) we obtain,

and

where again the first terms in the right hand side of equations (27) and (28) are the SUPG and PSPG contributions to the residual vector and the remaining terms define the approximations of and .

We do not assemble and store the resulting Jacobian, but rather we compute its action in the matrix-vector product needed in GMRES from element contributions, that is,

where A is the standard finite element assembly operator (Hughes, 2000), is the e-th contribution for the global Jacobian and x_e is the restriction of a general search direction to the element degrees of freedom. A similar strategy is employed in the successive substitution methods.

Test Problems

In this section we present the results obtained with the formulation described in the previous sections applied to two classical CFD problems. The first example consists of the lid driven cavity flow and the second is the flow over a backward facing step. For both examples we have tested the nonlinear algorithms proposed at Reynolds numbers 100, 500 and 1000. The numerical procedure considers a fully coupled u-p version of stabilized formulation using linear triangular elements. The computations were halted when the following criteria were met: < 10^–3 and < 10^–3 or the total number of nonlinear iterations exceeds 1000.

Lid Driven Cavity flow

The two-dimensional flow in a driven cavity which the top wall moves with a uniform velocity has been used rather extensively as a validation test case by many authors in the last years (see, Ghia et al, 1982). In this problem the Reynolds number is based on the size of the cavity, the flow velocity on the lid and fluid viscosity. The problem domain and mesh with 1,681 nodes and 3,200 elements are presented in Figs. 1a and 1b respectively.

Table 1 shows the results for tests with different nonlinear strategies to solve the lid driven cavity flow at Reynolds number 100, 500 and 1000. In all tests we employed GMRES(45) with nodal block-diagonal preconditioning to solve the linear problem. For the classical Newton-type methods the linear solver tolerance was set to 10^-6. For the inexact methods we have tested 0.1, 0.5 and 0.9 as the maximum linear solver tolerance, h_max. For each Table presented in this section, the first column lists the nonlinear method employed, where SS and ISS-h labels represent the Successive Substitution and Inexact Successive Substitution methods respectively and h is the maximum linear solver tolerance adopted by the inexact method. The second column shows the number of linear iterations (# LI) performed by GMRES. The third column gives the number of nonlinear iterations (# NLI) performed by each method. The fourth column presents the time spent in the solution process and the Euclidean solution norms (₂) were presented on the three last columns as solution quality indicators. Some Tables present two additional columns with the label (INFO) comprising warning messages or information about the solution process and (BCT) with the number of backtracking steps performed.

We can see in Table 1 that although the classical SS method requires less nonlinear iterations, it needed more GMRES iterations and thus, it is slower than the inexact method. We also observe that increasing the Reynolds number the solution process becomes more difficult.

Table 2 presents a performance comparison among the Newton-type implementations described in previous sections. These Newton-type methods differ by the form on which the Jacobian matrix evaluation is performed in the linearized problem. In the ISS method the nonlinear derivatives are not evaluated during the solution process. In the inexact Newton method (IN) the approximated Jacobian matrix is built and evaluated using the expressions in equations (26) to (28) from the start to the end of the nonlinear solution. We may also use a mixed solution strategy to circumvent initialization problems observed in some problems. In this strategy we enable the approximate Jacobian evaluation after k successive substitutions, thus this method will be labeled as k-ISS+IN. We have adopted arbitrarily in our tests k = 5 and h_max = 0.1.

Table 2

We can note in these Figures that for increasing Reynolds numbers convergence deteriorates. These Figures also show that convergence is faster for the methods based in the numerically approximated Jacobian evaluations than for the successive substitution methods.

In Table 3 we present the results of inexact nonlinear methods with backtracking enabled. Note that backtracking was invoked only for Re=500 and 1000. It was particularly important for recovering accuracy and reducing processing times of the IN solutions.

Figures 3 and 4 show the results obtained with the nonlinear methods for Reynolds numbers 100, 500 and 1000. These results were obtained with the inexact successive substitution method. Note in these Figures the vortex displacement for the center of the square for increasing Reynolds numbers.

Flow Over a Backward Facing Step

As a second example we consider the flow over a backward facing step, which also has become popular as a test problem for flow simulation codes. It consists of a fluid flowing into a straight channel which abruptly widens on one side. Numerical results obtained using a wide range of methods can be found in Gartling (1990). When the fluid flows downstream, it produces a recirculation zone on the lower channel wall, and for sufficiently high Reynolds it also produces a recirculation zone farther downstream on the upper wall. The finite element mesh with 1,800 elements and 1,021 nodes, boundary conditions and problem domain are present in Fig. 5a-b. Note that the Fig. 5a shows only the part of the computational domain that contains all the essential features.

Table 4 presents the performance results obtained for the Newton-type methods. The definitions adopted here are the same employed in the previous example. The additional column identified by RM, means one more quality parameter. It represents the Residual Mass found for the incompressibility constraint given by equation (2).

Here again, we observe that the inexact methods needed more nonlinear iterations. As in the previous example, these iterations required less time. Observe that for Reynolds 1000, only SS and ISS-0.1 methods were able to solve this problem for all variables. We note that the classical methods were more conservative, presenting less residual mass than the inexact methods. However, the residual mass of the converged inexact Newton solutions are of the same order of the required nonlinear accuracy. Table 5 shows the results obtained for the numerical Jacobian influence tests applied to the backward facing step problems. The tolerances and the other parameters were the same employed in the lid driven cavity flow problem.

Table 5

In Table 6 we present the results for the inexact methods with globalization procedures. Note that in this case the globalization procedures do not have any positive effect. For Re=500 and 1000 even the ISS method was unable to solve this problem with globalization switched on.

These Figures show an increasing difficulty to solve the backward facing step problem for high Reynolds numbers. For Reynolds 500 and 1000 the methods involving numerical approximate Jacobian stagnate without reaching convergence.

We can see the vortex recirculation forming on the top wall at high Reynolds numbers problems. This behavior is characteristic on backward facing step simulations.

Conclusions

In this work we compare several Newton-type algorithms to solve nonlinear problems arising from the SUPG/PSPG stabilized finite element formulation of the steady incompressible Navier-Stokes equations. Preconditioned GMRES is used as the linear iterative solver and the fully coupled Jacobian is numerically approximated by a truncated Taylor expansion. We introduced a mixed strategy which combines the successive substitution method with Newton's method using the numerically approximated Jacobians.

We observed that in general the inexact methods are simple to implement, fast and accurate. The numerically approximated Jacobian reduces the number of nonlinear iterations. However, the total number of GMRES iterations increases significantly, resulting in more CPU time than the corresponding successive substitution solutions. The mixed method combines the good features of both methods. In the backward facing step problem at Reynolds numbers 500 and 1000 the inexact solutions with the numerically approximated Jacobian needed globalization procedures to achieve a solution. This is an indication that effective solutions with this option may need more robust preconditioners than the simple nodal block-diagonal preconditioner used in this work. Our numerical results also indicate that globalization procedures must be used with care.

Sophisticated nonlinear solution methods are certainly needed for solving complex flow problems. However, they generally involve the choice of many parameters, which requires a high degree of expertise from the users. More numerical experiments are needed to access the influence of all parameters and to provide guidelines for the inexperienced user.

Acknowledgements

The authors would like to thank the financial support of the Petroleum National Agency (ANP, Brazil). The Center for Parallel Computations (NACAD) and the Laboratory of Computational Methods in Engineering (LAMCE) at the Federal University of Rio de Janeiro provided the computational resources for this research. We gratefully acknowledge Prof. T. E. Tezduyar from Rice University for his assistance with some details on the stabilized finite element formulation.

Paper accepted September, 2004. Technical Editor: Aristeu Silveira Neto.

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Publication Dates

• Publication in this collection
  07 Oct 2004
• Date of issue
  Sept 2004

History

• Accepted
  Sept 2004
• Received
  Sept 2004