Adjoint-consistent formulations of slip models for coupled electroosmotic flow systems
© Garg et al.; licensee Springer. 2014
Received: 6 February 2014
Accepted: 25 August 2014
Published: 27 September 2014
Models based on the Helmholtz `slip' approximation are often used for the simulation of electroosmotic flows. The objectives of this paper are to construct adjoint-consistent formulations of such models, and to develop adjoint-based numerical tools for adaptive mesh refinement and parameter sensitivity analysis.
We show that the direct formulation of the `slip' model is adjoint inconsistent, and leads to an ill-posed adjoint problem. We propose a modified formulation of the coupled `slip' model, which is shown to be well-posed, and therefore automatically adjoint-consistent.
Numerical examples are presented to illustrate the computation and use of the adjoint solution in two-dimensional microfluidics problems.
An adjoint-consistent formulation for Helmholtz `slip' models of electroosmotic flows has been proposed. This formulation provides adjoint solutions that can be reliably used for mesh refinement and sensitivity analysis.
KeywordsAdjoint techniques Electroosmosis Microfluidics Slip boundary conditions
Emerging micro- and nano-electromechanical systems (MEMS/NEMS) have a growing number of applications, ranging from lab-on-a-chip DNA analysis to micro-actuators . By scaling down processes, these systems offer savings in space, cost, and energy for scientific and technological advancement . However, the high manufacturing costs and complex architectures of these systems necessitate the use of numerical simulation tools for optimal design and precise control of their operation . Microfluidic devices operate over various length scales and are best described using multiphysics modeling that involves hydrodynamics, electroosmosis, and chemical species transport models. The development of accurate and efficient computational simulators of these devices is therefore challenging and resource intensive.
Numerical simulations of such complex engineering systems are typically targeted towards the calculation of specific Quantities of Interest (QoI) associated with the systems. Accurate estimation of local QoIs can be achieved using goal-oriented error estimation and adaptive techniques based on the use of adjoint methods ,. Adjoint methods can also be used to improve the computational performance of parameter sensitivity analyses , especially for systems with a large number of parameters. However, the application of adjoint methods to such coupled flow systems is an open area of study.
The objective of the current research work is to apply an adjoint-based Adaptive Finite Element Method (AFEM) to microfluidics problems for mesh refinement and parameter sensitivity analysis. There has been a growing interest in the modeling and numerical simulation of microfluidic systems -. A key issue that one faces while modeling and simulating microfluidic systems is the application of `slip' boundary conditions. Prachittham et al.  presented a space-time adaptive finite element method applied to an electroosmotic flow using large aspect ratio elements. However, their work did not consider adjoint techniques and did not use the `slip' boundary coupling condition. On the other hand, van Brummelen et al.  and Estep et al.  have shown the importance of the treatment of boundary flux coupling for the use of adjoint-based techniques. To our best knowledge, no advances in the application of adjoint-based techniques to microfluidics applications, particularly those involving `slip' boundary coupling, have yet been published in the literature.
In this work, we investigate coupled systems arising in electroosmotic flow (EOF). We show that a naive formulation of the `slip' model leads to an ill-posed adjoint problem and adjoint inconsistency . We provide numerical evidence that the corresponding adjoint solution exhibits spurious oscillations on a simple example dealing with a straight channel flow. Accordingly, we propose a modified variational formulation of the `slip' model, for which the adjoint problem is well-posed and can be computed and used in adaptive mesh refinement and parameter sensitivity analysis.
The paper is organized as follows: Section "Microfluidics modeling" describes the `slip' electroosmotic flow (EOF) model, with a brief discussion of the relevant physics and the applicability of such a model. Then, a variational formulation for a modified version of the `slip' model is presented and its adjoint is derived in Section "Variational formulation of the slip BC EOF model". An analysis of the ill-posed adjoint problem for the naive formulation is presented in Section "Ill-posedness of the adjoint problem for the naive formulation". Next, a variational formulation using penalty boundary conditions (henceforth called the penalty formulation) is proposed in Section "Penalty formulation of the slip BC EOF model". This new formulation is also employed to derive the corresponding adjoint problem and it is shown that the adjoint problem thus obtained is asymptotically consistent with the one derived in Section "Variational formulation of the slip BC EOF model". Numerical experiments are presented in Section "Results and discussion" that support the fact that the adjoint problem obtained using the penalty formulation is well-posed and that the adjoint solution is free of spurious artifacts. The adjoint obtained using the penalty formulation is used for goal-oriented adaptive mesh refinement and sensitivity analyses for a T-channel problem. Section "Conclusions" provides some concluding remarks, followed by a discussion of further work and future applications related to this class of coupled flow models. Finally, the appendix elaborates on the well-posedness of the modified formulation and presents relevant theoretical developments.
Microfluidics is the branch of fluid mechanics concerned with the understanding, modeling, and control of flows that occur on the micron scale, i.e. where the characteristic length (L) is of the order 10−6 m. Squires and Quake , and later Whitesides , have presented reviews of the physics and applications of microfluidics. Prominent among them are microflows driven by applied electric fields, through electroosmosis, electrophoresis, or both. In this paper, we consider electroosmotic flow devices, which find wide use in commercial and industrial applications ,. These devices utilize the properties of the electric double layer (also called the Debye layer) to drive a bulk fluid flow. More detailed descriptions of the electric double layer and electroosmotically driven microfluidic devices can be found in the microfluidics literature ,,.
where E denotes the applied electric field, and a new parameter λ=є Ψ0/μ has been introduced. Here, є is the permitivity (or dielectric constant) of the fluid medium. A detailed derivation of this approximation can be found in a standard reference . The validity of this approximation has been verified for several examples through both experiments and numerical simulations .
We see that the slip model spares one from solving the Poisson-Boltzmann equation, whose solution exhibits a thin layer near the wall. As a remark, the slip boundary approximation model given by Eq. (6) and (7) is widely used throughout the microfluidics research and development community for modeling and simulation . The model is even included in the commercial Finite Element software package COMSOL Multiphysics .
where we have denoted the normal (∇φ•n) and tangential (∇φ•t) derivatives of φ by ∂ n φ and ∂ t φ, respectively. This coupling is equivalent to the one given by Eq. (7c). We now proceed to derive the weak formulation for Eq. (6) using the modified coupling given by Eq. (8).
Variational formulation of the slip BC EOF model
Variational formulation of primal problem
We have the following proposition.
where c(Ω) is a positive constant that depends only on the domain Ω.
Therefore, , and an application of the Lax-Milgram theorem gives the well-posedness of the variational problem, and the bound . □
where U = (φ,w,p) and V = (ψ,v,q).
We have thus incorporated the coupling condition within our bilinear form and can prove that the coupled problem (26) is well-posed. We refer the interested reader to Appendix A and proceed with the derivation of the corresponding adjoint problem.
Note that the adjoint Stokes problem solution w* and p* are coupled to the adjoint potential φ* through the lift operator ℓ(.) acting on the test function ψ. The adjoint problem (28) is also well-posed, see Corollary 2 in Appendix A.
We readily observe that the adjoint Stokes problem can be solved first, independently of the adjoint potential problem, but that the latter does depend on the former through the Neumann coupling condition Eq. (37a). We also note that this coupling condition involves the tangential derivatives of the adjoint stress tensor on the boundary.
Ill-posedness of the adjoint problem for the naive formulation
Thus the adjoint problem for the ill-posed primal problem contains four boundary conditions, despite there being only three variables. Therefore, the ill-posed primal problem leads to an adjoint inconsistent formulation, specifically in the boundary terms. Such an adjoint inconsistency can lead to oscillations in the discrete solutions of the adjoint problem .
Penalty formulation of the slip BC EOF model
Penalty formulation of the primal problem
One can observe that the penalty method replaces the Dirichlet boundary conditions with a Robin condition. However, upon taking the limit ∈→0 one formally recovers the original problems Eq. (9) and Eq. (14).
Adjoint problem associated with the penalty formulation
In the next section, we show that above problem is asymptotically consistent with the previous formulation of the adjoint problem, in the sense that we recover the adjoint corresponding to the strong problem, i.e. Eq. (35), Eq. (36), and Eq. (37) as ∈ tends to zero.
Consistency of the adjoint penalty problem
which are the same as those for the non-penalized adjoint in the limit ∈→0. Eq. 53d corresponds to a penalty representation of the tangential boundary flux. Further discussion of this representation is presented in . We thus see that the penalized formulation of the electroosmotic flow problem is adjoint consistent in the limit as ∈→0, and the use of the discrete penalized adjoint solution in adjoint-based error estimation and sensitivity analysis is justified. Thus, if the forward problem is computed numerically using a discrete representation of Eq. (39), the adjoint solution can also be easily computed by taking the transpose of the stiffness matrix associated with the forward problem.
Results and discussion
We now consider the application of the new EOF formulation on specific microfluidic examples. First, we simulate a flow in a straight microchannel driven purely by electroosmosis. The objective here is to highlight the convergence and stability properties of the adjoint solution. We then showcase an adjoint-based adaptive strategy for mesh refinement on a T-shaped microchannel flow and adjoint-based parameter sensitivity analyses. We discuss the improvement of the convergence rates with respect to quantities of interest and their sensitivities when using adjoint-based techniques. Simulations are performed using the adjoint capabilities added to the libMeshlibMesh Finite Element library . For both applications, second-order Lagrange elements are employed for the potential and velocity approximations. Linear Lagrange elements are selected to approximate the pressure field in order to satisfy the inf-sup condition. Initial meshes in all the experiments dealing with the straight and T-channel domains consist of structured meshes of bi-quadratic quadrilateral elements. Numerical errors to generate the convergence plots are estimated in this work using so-called overkilled reference solutions of the two problems. These are obtained on a uniform mesh of 428,676 degrees of freedom for the straight channel problem and a combined adaptive-uniform mesh with 288,160 degrees of freedom for the T-channel problems. Numerical solutions are calculated using an ILU preconditioned GMRES iterative method for both problems. The linear algebra library PETSc is accessed through libMeshlibMesh to obtain these solutions. The penalty parameter ∈ was set to the constant value of 10−8 for all the numerical experiments.
Electroosmotic flow in a straight channel
Numerical experiments are performed here in the case of an electroosmotic flow in a straight channel. The channel has unit width and the length is five times the width. Since the objective of these simulations is to illustrate the numerical properties of the adjoint solution obtained by using Eq. (39), we set arbitrary values of the model parameters rather than choosing values representative of an actual flow. The fluid viscosity μ, electroosmotic slip parameter κ, and fluid density ρ are all taken to be unity. Constant potentials Φ i =8 and Φ o =0 are prescribed at the inlet Γin and outlet Γout boundaries, respectively. The electric conductivity of the fluid is chosen as σ c =1+x (note that this particular form of the conductivity is chosen for no other reason than better illustrate the properties of the computed adjoint).
Electroosmotic flow in a T-channel
Crossing T- and H-channels are commonly utilized in microfluidics. Applications typically involve mixing of two chemical species , purification , or fluid identification . However, numerical modeling of electroosmotic flows with slip boundary conditions in such geometrical configurations poses distinctive challenges due to the presence of corner singularities . One immediate consequence is the observation of reduced convergence rates in the approximation of the global solution. A possible remedy is to use adaptive finite element methods to help restore the optimal convergence properties of such singular problems . Likewise, adaptive methods can also improve the convergence behavior of the adjoint solution and potentially restore the optimal rates that one may expect when estimating linear QoIs.
Values of the input parameters in the case of the T-channel flow
Estimated reference values of QoI and of its sensitivity to Φ i , Φ o , and κ
d Q/d Φ o
d Q/d Φ i
d Q/d λ
We also used an adjoint method to compute parameter sensitivities for the given QoI to the parameters Φ i , Φ o , and λ. The advantage of using an adjoint method for sensitivity analysis is that the sensitivity to all three parameters could be found with a single adjoint solve. This is considerably more efficient than using a finite difference or a forward sensitivity method. In addition, we can also combine the adjoint-based mesh refinement and sensitivity analysis for further improvements in the convergence of the sensitivities.
In fact, on account of the geometric corner singularities present in the problem, we obtain an inferior convergence rate on using uniform refinement. However, with the adaptive method we obtain a rate of 1.5 (vs dofs) for the QoI, which can be said to be semi-optimal. We had observed earlier that there is a loss of one order in the convergence rate for the forward velocity and adjoint potential for the straight channel problem where there are no corner singularities. We recall that with second-order Lagrange Finite Elements this would result in a convergence rate of 1.5 () for a linear QoI.
We have presented an analysis of an electroosmotic flow model with slip boundary conditions and its adjoint. The slip boundary conditions require the evaluation of potential derivatives on the boundary, which increases the regularity requirements on the potential. We emphasize that a naive enforcement of the standard slip boundary condition leads to an ill-posed adjoint problem (see Section "Ill-posedness of the adjoint problem for the naive formulation"). This leads to instabilities in the computed adjoint, illustrated by numerical experiments in Section "Results and discussion". A well-posed adjoint problem can be obtained by modifying the slip boundary condition (u+λ∇Φ=0), i.e. specifying the normal velocity at the wall independently of the potential (u•n=0,u•t+λ∇Φ•t=0).
We further proposed a penalty formulation of the forward problem that requires no extra regularity for the potential, and leads to a well-posed, asymptotically consistent adjoint formulation as well. The penalty boundary conditions lead to a weak enforcement of the boundary coupling, allowing us to easily compute the adjoint problem using the adjoint capabilities of libMeshlibMesh.
Finally, we presented numerical experiments for a simple straight channel microflow and a more challenging T-channel flow. The convergence results for the straight channel problem indicate that the primal velocity and the adjoint potential converge at sub-optimal rates due to the nature of the coupling between the potential and the velocity. For the T-channel, we presented QoI computation and QoI adjoint sensitivity results for a practical engineering QoI. We observed a loss of convergence order due to the singularities in the T-channel geometry, and substantial improvements in the rate on using an adjoint-based adaptive method. However, the fully optimal convergence rate for the QoI could not be achieved, possibly due to the convergence properties of the adjoint potential.
Future work will involve the application of more sophisticated adjoint-based error estimators to further improve the convergence properties of the approximate solutions and obtain reliable a posteriori error estimates for complex applications.
Appendix A: Well-posedness of coupled formulation
where , , and and continuous bilinear forms, and and are continuous linear forms.
If and satisfy the inf-sup conditions on Z×Z and on X×X, respectively, then the above problem can be solved sequentially: First solve (58a) for Φ∈Z. Then, since is a continuous linear form on X, Eq. (58b) can be solved for u∈X.
The fact that one-way coupled problems (with bounded coupling) are well-posed is probably well known. We now provide an extension of this result for the aggregated bilinear form .
Let denote the aggregated bilinear form:
then satisfies the inf–sup conditions:
The proof is similar to the Brezzi–Babuška equivalence theorem; see e.g. , Proposition 2.36].
We first prove (60a). Using inf-sup stability of we obtain:
From (58b), inf-sup stability of , and continuity of we obtain:
Finally, summing both contributions,
To prove (60b), let (Ψ,v)∈Z×X such that
Choosing (Φ,u)=(0,u), we obtain
which upon invoking (59d) yields v=0. Next, choosing (Φ,u)=(Φ,0), we obtain
which upon invoking (59b) yields Φ=0.
where is the inf-sup constant for the bilinear form , λ as required by Proposition 1 and c(Ω) is as given by Eq. (19). □
while is simply the null map. This easily gives the a-priori bounds on φ and (w,p). □
The inf-sup stability of , and Eq. (73a) and (73b) then easily give the a priori error estimates. □
Vikram Garg is grateful for the support of the Bruton fellowship and the University Continuing fellowship from The University of Texas at Austin. Kris van der Zee is grateful for the support of this work by the 2010 NWO Innovational Research Incentives Scheme (IRIS) Grant 639.031.033. Serge Prudhomme is sponsored by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Serge Prudhomme is also a participant of the KAUST SRI center for Uncertainty Quantification in Computational Science and Engineering.
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