FEM simulation of thixo-viscoplastic flow problems: Error analysis

This note is concerned with error analysis of FEM approximations for quasi-Newtonian modelling of thixo-viscoplastic, TVP, ﬂow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efﬁcient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization parameter, nonoptimal estimate for microstructure, and extra regularization requirement for velocity, is due to weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with higher order stabilized FEM. Furthermore, we showed the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical simulations of TVP ﬂow problems in a 4:1 contraction conﬁguration.


Introduction
FEM approximation of thixo-viscoplastic flow problems using quasi-Newtonian modelling approach is a straightforward way to generalize the FEM standard setting of Navier-Stokes equations, as well standing tool for simulating incompressible flow problems [9].In this context, the extended viscosity, µ(•, •), is dependent on internal material microstructure parameter, λ, beside the shear rate, ||D||, for the generalized Navier-Stokes equations [10].The well defined approximation for the term ||D|| −1 , as for instance Papanastasiuo approximation [11], is used to deal with the singularity of the modelling, D II = 1 2 D(u) : D(u) denotes the second invariant of the strain rate tensor, and k is the regularization parameter.Then, the viscosity in generalized Navier-Stokes equations is given as follows µ(D II,r , λ) = η(D II , λ) + τ (λ) and the full set of equations for thixo-viscoplastic problems reads in Ω with external forces f u , and f λ .u, p, and λ denote velocity, pressure, and structure parameter, respectively.The supplemented evolution equation for the structure parameter to generalized Navier-Stokes equations in (3) induces the timedependent process of competition between the breakdown, G, and the buildup, F, inhabited in the material.A collection of thixotropic models with various choices of η, τ , F and G is given in Table 1.We briefly define the thixotropic model as The paper is organized as follows.In section §2, we show the wellposedness of the continuous problem followed by the best approximation for the discrete one.Next in section §3, we present the numerical simulations of TVP flow problems in a 4:1 curved contraction configuration showing the importance of not ignoring the thixotropy inhabited in material.In summary section §4, we outline the effect of weak coercivity of microstructure operator and regularization parameter on energy inequality, beside the importance of taking into account the thixotropy inhabited in material giving a way to higher order stabilization FEM and better understanding of flow characteristics.
Here η 0 and τ 0 are initial plastic viscosity and yield stress, respectively, in the absence of any thixotropic phenomena.η ∞ and τ ∞ are thixotropic plastic viscosity and yield stress.Λ c is the critical elastic strain, and G 0 is the elastic modulus of unyielded material.M a and M b are buildup and breakage constants, and g, p, m, n are rate indices.

Finite element approximations
For finite element approximations, we start by deriving the variational form for thixo-viscoplastic flow problems, followed by the wellposedness results of the continuous problem, then we show the best approximation for the discrete problem.Let's consider the spaces 2 , W := T×V, and Q := L 2 0 (Ω) associated with the corresponding norms H 1 -norm ||•|| 1 and L 2 -norm ||•|| 0 , respectively, and T , V and W := T × V their corresponding dual spaces [1].We set ũ = (λ, u), ṽ = (ξ, v), and define on W × W The weak formulation for the thixo-viscoplastic flow problems (3) reads: where operators •), and l(•) are given as follows The wellposedness results are stated in theorem 2.1.
If the body force in the pressure bound (12) tends towards zero, the limit for pressure is not necessarily zero due to regularization parameter, means that the pressure is underdetermined in the rigid zone.Moreover, the high order regularity of microstructure parameter is not controlled, i.e. it is only bounded with L 2 -norm and boundary norm (13).
The approximation of TVP problem, in its general abstract form using conforming framework, is to seek an approximated solution where, T h ⊂ T, V h ⊂ V, W h ⊂ W, and Q h ⊂ Q are finite dimensional subspaces with the superscript h being a parameter dependent on the mesh spacing.The problems that we have to solve here are the existence and uniqueness of the solution Copyright line will be provided by the publisher (ũ h , p h ) and the estimation ||λ − λ h || 0 , ||u − u h || 0 , and ||p − p h || 0 .We assume that the inf − sup condition for the pair where β is independent of h.Clearly, the inclusion ker B h ⊂ ker B is not true in general.Nevertheless, results of theorem 2.1 concerning existence, uniqueness, and boundedness of the solution with the data are directly applied here.Indeed, the necessary properties of a ũ( ũ)(•, •) are satisfied in whole space W. In addition, we assume the following conditions Proposition 2.3 (Constrained monotonicity) Let u be the solution of TVP problem, for all ũ = (λ, u), ṽ = (ξ, v) ∈ W 0 , we have where, (ζ, η) = (λ − ξ, u − v).Theorem 2.4 Let f u ∈ (L 2 (Ω)) 2 and f λ ∈ L 2 (Ω), the approximate thixo-viscoplastic problem (14) has a unique solution (ũ h , p h ) = (λ h , u h , p h ) ∈ W h × Q h .Assume in addition that the solution satisfies the conditions (16) and (17).Then, the approximation solution satisfies the following best approximation where Cλ,λ , Cλ,u , Cu,u , and Cu,λ are a constants depending only on Remark 2.5 The energy inequality for the microstructure (22) states that the error for the approximation of microstructure in L 2 -norm is bounded by the error of best approximation of the solution in H 1 -norm, which is not optimal.In contrast, the energy inequality (23) state that the error estimate for velocity approximation in H 1 -norm is bounded by the best approximation of the solution in H 1 -norm as well which is optimal modulo the regularity requirement.These coarseness, i.e. the extra regularity requirement for velocity on one hand and the non-optimality of the estimate for microstructure on the other hand, is due to the weak coercivity of a λ (•)(•, •) i.e. coercivity only in L 2 -norm and boundary norm.P r o o f.To derive the error we subtract the approximated TVP (14) problem from the exact (6) TVP problem ), be test functions and add respectively on both side of (24) and (25) terms a λ (ṽ We apply monotonicity and continuity of a λ (•)(•, •) and a u (•)(•, •) on LHS and RHS of ( 26) and ( 27), respectively, to have, and to conclude where C λ,λ , C λ,u , C u,λ , and C u,u are given as follows where Cλ,λ , Cλ,u , Cu,λ , and Cu,u are dependent on C λ,λ , C λ,u , C u,λ , and C u,u .Thus, using triangular inequalities we conclude the proof.
The finite element approximations of the problem ( 6) have to take care of its saddle point character, due to the bilinear form (9), the weak coercivity of a λ (•)(•, •), and the dependency of solution on regularization parameter k.We opt for higher order stable FEM pair biquadratic for velocity and piecewise linear discontinuous for the pressure, Q 2 /P disc 1 , and higher order quadratic for structure parameter Q 2 with the appropriate stabilization terms [10,12].On the one hand, the higher order choice for velocity counterbalances the regularization impact and stabilization on the other hand enhances the coercivity to match the complete norm of the microstructure space T equivalently as H 1 -norm i.e.
where j λ (ξ, ξ) is bilinear form supplementing the microstructure equation.Indeed, let the domain Ω be partitioned by a grid with K ∈ T h which are assumed to be quadrilaterals such that Ω = k∈T h K .For an element K ∈ T h , we denote by E(K) the set of all 1-dimensional edges of K. Let E i := k∈T h E(K) be the set of all interior element edges of the grid T h .We define the conforming finite element spaces The stabilized approximate problem reads: The stabilization term j ũ(•, •) is given as follows [12] The stabilization ( 47) is consistent and it is expected to recover the optimal order of convergence.The detailed corresponding analysis goes beyond the goal of this note and will be reported in a separate work.

Numerical simulations
We investigate numerical solutions of Houska's [6] thixo-viscoplastic material in a 4:1 curved contraction configuration.The fully-developed flow conditions according to Houska thixotropic model are imposed at entry, Γ − , together with no-slip on walls of reservoir, Γ.
The numerical solutions are obtained using a monolithic Newton-multigrid FEM solver.On the one hand, we are using a global adaptive discrete Newton method to linearize the discrete non-linear TVP problem, where the adaptive discrete Newton method is based on step-length in divided difference.The adaptive strategy is exclusively due to the current convergence rate of residual.On the other hand, the linearized systems inside the outer Newton loops are solved using a coupled geometrical multigrid solver based on local pressure Schur complement (LPSC) schemes.They are simple iterative relaxation methods solving directly on element level and performing an outer block Gauss-Seidel iteration.The local character of this procedure together with a global defect-correction mechanism on one hand, and the choice of discontinuous FE approximations for pressure (P disc 1 ) on the other hand, results in an efficient solver for TVP problems.For details, we refer to [2][3][4].Our emphasis is to revisit the flow characteristics by not ignoring the thixotropy inhabited in a material in pipelines, which is a typical industrial application in transportation of waxy crude oils.Figure 1 illustrates the impact of breakdown parameter M b on the flow in the vicinity of walls.By a simple increase in the breakdown parameter, we induce more breakdown layers close to walls of downstream section.Clearly, more breakdown layers prevent the material from resting along pipelines circumventing the need for extra lubrication and regularizing the restart pressure to optimal settings.

Summary
We investigated the essential part of error analysis of FEM approximations for the quasi-Newtonian modelling approach of thixo-viscoplastic flow problems.In this regard, the standard FEM settings of Navier-Stokes equations are adapted to deal with the new thixo-viscoplastic generalized Navier-Stokes equations.The wellposedness results beside the boundedness of the solutions with the data are used to set a constrained monotonicity of the coupled problem, which serves beside the continuity to elegantly elaborate the energy inequality of the best approximation.On the one hand, the energy inequality for the microstructure shows that the estimation of the error for the approximation of microstructure in zero norm is bounded by the error of best approximation of the solution in one norm, causing the loss of one order of convergence.On the other hand, the energy inequality for velocity shows that the error estimate for velocity approximation in one norm is bounded by the error estimate of the best approximation of the solution in one norm modulo an extra regularity requirement which is a clear manifestation of the weak coercivity of microstructure parameter operator in zero norm and boundary norm only.Moreover, constants in energy inequalities are proportionally dependent on the regularization parameter.We dealt with these coarseness, the proportional dependency of constants on the regularization parameter and the weak coercivity of microstructure operator, by opting for stabilized higher order FEM.The higher order FEM choice counterbalances the regularization effect, while the stabilization enhances the coercivity to an equivalent one norm.
We analysed numerically solutions of Houska's [6] thixo-viscoplastic material in a 4:1 curved contraction configuration using monolithic Newton-multigrid FEM solver.We investigated the impact of thixotropy breakdown parameter M b on material microstructuring level λ.In fact, increasing the breakdown parameter induces more breakdown layers in vicinity of walls of downstream channel preventing the material from rest along pipelines, which circumvents the need for extra lubrication and optimizes the restart pressure settings.

Table 1 :
Thixotropic models (6) continuity constant of thixotropy build-up tri-linear form due to Hölder inequality (L ∞ , L 2 , L 2 ).Now, we move to essential part of finite element approximation of comparing the discrete solution (λ h , u h , p h ) of the approximated TVP problem (14) to the exact solution (λ, u, p) of the continuous TVP problem(6).The straightforward way is to use monotonicity combined with the continuity for the coupled operator a ũ(•)(•, •) which is not true in this case.So, we use a constrained monotonicity Proposition (2.3) together with the continuity Proposition (2.2) to establish our results.Proposition 2.2 (Continuity)