Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.
Citation: Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows[J]. Electronic Research Archive, 2021, 29(4): 2755-2770. doi: 10.3934/era.2021012
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We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.
The numerical simulation of viscoelastic fluid flows is challenging due to the strong coupling of governing equations and the large algebraic system with multiple dependent variables. Viscoelastic behavior of the fluid is often represented by a nonlinear constitutive model in the form of hyperbolic partial differential equations, which requires a stabilization technique for finite element approximations such as the streamline upwinding Petrov–Galerkin (SUPG) method and the discontinuous Galerkin method. In addition, finite element spaces for the velocity, pressure, and stress should satisfy the inf-sup condition if the standard mixed method is applied for numerical simulations. To overcome such difficulties, least-squares (LS) methods are frequently used to simulate viscoelastic flow problems [4,6,9,14] because they provide the flexibility of choosing finite element spaces and no additional stabilization is needed. In this work we expand on the LS method [14] by considering a nonlinear a posteriori error estimator of residual type for the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid.
The PTT model is a popular viscoelastic model in polymeric fluid simulations [17,16], where the extra-stress is written as a superposition of the polymeric and viscous stresses. The linear PTT model was proposed by Phan-Thien and Tanner in 1977, and Phan-Thien subsequently proposed the exponential PTT model in 1978.
A posteriori error estimators play an important role in measuring the accuracy of numerical solutions and can be used to guide adaptive refinements. They are also effective and reliable for error control [1,4,15]. Previous works on the Stokes equations and the linear PTT model [11,14] show that an a posteriori error estimator serves as an indicator to adjust mass conservation weights on LS approaches to solve flows past a transverse slot. However, the numerical computations have been performed on uniform grids with more degrees of freedom. In [13], the results also show that grid effects of the LS method can not be reduced on some uniform refined grids, and geometric discontinuities cause corner singularities for flows in complex geometries.
On the basis of these studies, we consider an a posteriori error estimator for the exponential PTT model to adjust mass conservation weights and to develop mesh refinement criteria on the LS approach. Furthermore, we estimate the coercivity and continuity for the homogeneous LS functional, which involves the sum of the linearized equation residuals measured in the
The organization of this paper is as follows. Section 2 presents the exponential PTT flow model and the LS functional for the Newton linearized system. In Section 3 the least squares problem is defined and the coercivity and continuity properties of the LS functional are proved. Section 4 presents an a priori error estimate for the LS approximations and introduces an a posteriori error estimator. Section 5 provides numerical results for numerical examples, and finally conclusions follow in Section 6.
The PTT model is one of widely used viscoelastic models in polymeric fluid simulations, where the extra-stress is written as a superposition of the polymeric stress
σeϵληptr(σ)+λ(u⋅∇σ−(∇u)T⋅σ−σ⋅∇u)=2ηpD. | (1) |
In (1)
eϵληptr(σ)=1+ϵληptr(σ)+12(ϵληp)2(tr(σ))2+O((ϵληp)3), |
hence,
eϵληptr(σ)≈1+ϵληptr(σ)+a2(ϵληp)2(tr(σ))2. | (2) |
The constitutive equation (1) with
Consider an incompressible exponential PTT viscoelastic fluid flow in
G1U:=Re(u⋅∇u)−∇⋅σ−∇⋅τ+∇p, | (3) |
G2U:=∇⋅u, | (4) |
G3U:=τ−2βD, | (5) |
G4U:=We(u⋅∇σ−(∇u)T⋅σ−σ⋅(∇u))+σ˜e(σ)−2(1−β)D, | (6) |
where
˜e(σ):=1+ϵWe1−βtr(σ)+a2(ϵWe1−β)2(tr(σ))2. |
Collecting the constitutive equation (1) with the substitution (2) and by nondimensionalizaing, we have governing equations of the viscoelastic flow represented by the boundary value problem:
GU=FinΩ, | (7) |
u=0 onΓ, | (8) |
where
We assume that the scalar pressure
Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
To define the LS functional, we consider Newton linearization of the nonlinear equations (7)-(8) about known approximation
LU=F inΩ, | (9) |
u=0 onΓ, | (10) |
where
L1U:=Re(un⋅∇u+u⋅∇un)−∇⋅σ−∇⋅τ+∇p, | (11) |
L2U:=∇⋅u, | (12) |
L3U:=τ−2βD, | (13) |
L4U:=L4,1U+L4,2U+L4,3U+L4,4U, | (14) |
and
A(∇u,σ)=(∇u)Tσ+σ(∇u),L4,1U:=σ−2(1−β)D,L4,2U:=We(u⋅∇σn+un⋅∇σ−A(∇un,σ)−A(∇u,σn)),L4,3U:=(ϵWe1−β)(tr(σn)σ+tr(σ)σn),L4,4U:=a2(ϵWe1−β)2[(tr(σn)tr(σn))σ+2(tr(σn)tr(σ))σn], |
and
f1:=f+Re(un⋅∇un),f2:=0f3:=0f4:=We(un⋅∇σn+A(∇un,σn))+(ϵWe1−β)(tr(σn)σn)+a2(ϵWe1−β)2(tr(σn)tr(σn))σn. |
The LS functional for (9)-(10) is then defined by
J(U;F)=4∑j=1∫ΩWj(LjU−fj)2dΩ. | (15) |
The weights of positive constants
Remark 1. Since the system of governing equations (7)-(8) is nonlinear, the linear functional (15) is minimized on each Newton iteration with
We consider the product space
Φ:=V×Q×Σs×Σ:=H1(Ω)2×L2(Ω)×L2(Ω)2×L2(Ω)2. |
The least-squares minimization problem for the solution of system (9)-(10) is to choose
J(U;F)≤J(V;F) ∀V∈Φ, | (16) |
i.e., seek
To prove the coercivity and continuity of
J0:=3∑j=1Wi‖LjU‖20+W4‖L4,1U+L4,2U+L4,3U‖20. | (17) |
Here, the norm
Theorem 3.1. Suppose the known approximations
M:=max{‖un‖∞, ‖∇un‖∞,‖σn‖∞, ‖∇σn‖∞}<∞. | (18) |
Then, for any
c0(‖u‖21+‖p‖20+‖τ‖20+‖σ‖20)≤J(U;0) |
and
J(U;0)≤c1(Re2‖u‖21+‖p‖21+‖τ‖21+‖σ‖21) |
if
Proof. For a lower bound of
‖L4,4U‖0=a2(ϵWe1−β)2(‖(tr(σn)tr(σn))σ‖0+2‖(tr(σn)tr(σ))σn‖0)≤3a(ϵWe1−β)2‖σn‖20‖σ‖0≤3a(ϵWe1−β)2M2‖σ‖0, | (19) |
and using the inequality
J(U;0)≥12J0−W4‖L4,4U‖20≥12J0−3aW4(ϵWe1−β)2M2‖σ‖0. | (20) |
It was proved in [14] that there exists a constant
J0≥¯C0(‖u‖21+‖p‖20+‖τ‖20+‖σ‖20) |
if
J(U;0)≥¯C02(‖u‖21+‖p‖20+‖τ‖20)+(¯C02−¯C04)‖σ‖20≥c0(‖u‖21+‖p‖20+‖τ‖20+‖σ‖20), | (21) |
where
For the finite element approximation of (7)-(8), we assume that the domain
Vh={vh∣vh∈V∩(C0(Ω))2, vh∣e ∈Pr+1(e)2 ∀e∈Th},Qh={qh∣qh∈Q∩C0(Ω), qh∣e ∈Pr+1(e) ∀e∈Th},Σhs={ζh∣ζh ∈Σs∩(C0(Ω))2×2, ζh∣e ∈Pr+1(e)2×2 ∀e∈Th},Σh={ςh∣ςh ∈Σ∩(C0(Ω))2×2, ςh∣e ∈Pr+1(e)2×2 ∀e∈Th}. |
Let
infuh∈Sh‖u−uh‖l≤Chm‖u‖m+l∀u∈Hm+l(Ω), | (22) |
for
The discrete minimization problem for (16) is to choose
J(Uh;F)=infVh∈ΦhJ(Vh;F), | (23) |
where
B(Uh,Vh)=F(Vh), ∀Vh∈Φ, | (24) |
where
B(Uh,Vh):=(LUh,LVh), | (25) |
F(Vh):=(F,LVh). | (26) |
The following error estimate for the solution of (23) is obtained by the standard way, using the coercivity and continuity properties of the functional
Theorem 4.1. Consider approximating the solution to (9)-(10), where the known functions
‖u−uh‖1≤Chm(‖τ‖m+1+‖σ‖m+1+‖p‖m+1+Re‖u‖m+1), | (27) |
‖p−ph‖0≤Chm(‖τ‖m+1+‖σ‖m+1+‖p‖m+1+Re‖u‖m+1), | (28) |
‖τ−τh‖0≤Chm(‖τ‖m+1+‖σ‖m+1+‖p‖m+1+Re‖u‖m+1), | (29) |
‖σ−σh‖0≤Chm(‖τ‖m+1+‖σ‖m+1+‖p‖m+1+Re‖u‖m+1), | (30) |
for
Proof. By Theorem 3.1 and the approximation properties in (22), we have
‖u−uh‖1≤(‖u−uh‖21+‖p−ph‖21+‖τ−τh‖21+‖σ−σh‖21)1/2≤infVh∈Φhc1c0(Re2‖u−vh‖21+‖p−qh‖21+‖τ−ζh‖21+‖σ−ςh‖21)1/2≤Chm(‖τ‖m+1+‖σ‖m+1+‖p‖m+1+Re‖u‖m+1). | (31) |
By the same approach, we can obtain the desired estimates for
Note that we obtain the error bounds
For the residual of the first-order system (7)-(8), consider a nonlinear a posteriori error estimator of the following form:
g=∑e∈Thge,wherege=4∑j=1‖GjUh−F‖20,e | (32) |
and
Theorems 3.1 and 4.1 require the small
We consider two numerical examples. The first problem is chosen for convergence tests with the known exact solution in the unit square domain, and the second is a slot flow problem. All variables are approximated by
Consider the flow in a planar channel on the domain
First, in order to appropriately adjust the weight
Figure 2 shows the errors of LS solutions with
Consider the PTT flow past a slot in a channel for
We consider the LS method (15) with the adaptively refined algorithms, which we name it the ALS method. This refinement technique is similar to the approach used for the Carreau model in [10]. For the mesh refinement strategy, we apply the grading function
f(ϕe)=|e|max−(ϕe−ϕemin)Δ|e|Δϕe, |
where
Mesh | Type | Method | |||
Mesh S | Initial uniform grids | LS | 348 | 4 | – |
Mesh T | Initial uniform grids | LS | 1536 | 4 | – |
Mesh U | Uniform refined grids with Mesh T | LS | 24576 | 5 | 2 |
Mesh G | Adaptive grids by |
ALS | 12993 | 4 | 3 |
Mesh H | Adaptive grids by |
ALS | 18320 | 4 | 5 |
aNk represents the number of elements at the k refinement step. b S is the number of Newton steps for convergence. c k is the number of mesh refinements. |
The optimal weight
Grid independence results of
Furthermore, in Figure 9, we present contours of
Finally, we compare streamlines of the ALS solutions for the exponential PTT (
We considered a nonlinear a posteriori error estimator based on a first-order LS method for the exponential PTT viscoelastic model of strong flows. The LS method was developed by assigning proper weights to the terms of the LS functional, in which the mass conservation weights are obtained by the a posteriori error estimator. We also proposed an adaptive LS (ALS) for the exponential PTT model, where the a posteriori error estimator of the LS solution is used for adaptive mesh refinements. The proposed approach resolves some difficulties related to the presence of corner singularities in this second example. In addition, the ALS approach with adaptive mesh refinements improves computational efficiency.
1. | Hsueh-Chen Lee, A least-squares finite element method for steady flows across an unconfined square cylinder placed symmetrically in a plane channel, 2021, 504, 0022247X, 125426, 10.1016/j.jmaa.2021.125426 | |
2. | Hsueh-Chen Lee, Hyesuk Lee, Equal Lower-order Finite Elements of Least-squares Type in Biot Poroelasticity Modeling, 2023, -1, 1027-5487, 10.11650/tjm/230702 |
Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
Mesh | Type | Method | |||
Mesh S | Initial uniform grids | LS | 348 | 4 | – |
Mesh T | Initial uniform grids | LS | 1536 | 4 | – |
Mesh U | Uniform refined grids with Mesh T | LS | 24576 | 5 | 2 |
Mesh G | Adaptive grids by |
ALS | 12993 | 4 | 3 |
Mesh H | Adaptive grids by |
ALS | 18320 | 4 | 5 |
aNk represents the number of elements at the k refinement step. b S is the number of Newton steps for convergence. c k is the number of mesh refinements. |
Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
Mesh | Type | Method | |||
Mesh S | Initial uniform grids | LS | 348 | 4 | – |
Mesh T | Initial uniform grids | LS | 1536 | 4 | – |
Mesh U | Uniform refined grids with Mesh T | LS | 24576 | 5 | 2 |
Mesh G | Adaptive grids by |
ALS | 12993 | 4 | 3 |
Mesh H | Adaptive grids by |
ALS | 18320 | 4 | 5 |
aNk represents the number of elements at the k refinement step. b S is the number of Newton steps for convergence. c k is the number of mesh refinements. |