Christopher Linsenmann

An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method

The immersed boundary method (IB) is known as a powerful technique for the numerical solution of fluid-structure interaction problems as, for instance, the motion and deformation of viscoelastic bodies immersed in an external flow. It is based on the treatment of the flow equations within an Eulerian framework and of the equations of motion of the immersed bodies with respect to a Lagrangian coordinate system including interaction equations providing the transfer between both frames. The classical IB uses finite differences, but the IBM can be set up within a finite element approach in the spatial variables as well (FE-IB). The discretization in time usually relies on the Backward Euler (BE) method for the semidiscretized flow equations and the Forward Euler (FE) method for the equations of motion of the immersed bodies. The BE/FE FE-IB is subject to a CFL-type condition, whereas the fully implicit BE/BE FE-IB is unconditionally stable. The latter one can be solved numerically by Newton-type methods whose convergence properties are dictated by an appropriate choice of the time step size, in particular, if one is faced with sudden changes in the total energy of the system. In this paper, taking advantage of the well developed affine covariant convergence theory for Newton-type methods, we study a predictor-corrector continuation strategy in time with an adaptive choice of the continuation steplength. The feasibility of the approach and its superiority to BE/FE FE-IB is illustrated by two representative numerical examples.

Numerical simulation of surface acoustic wave actuated cell sorting

We consider the mathematical modeling and numerical simulation of high throughput sorting of two different types of biological cells (type I and type II) by a biomedical micro-electro-mechanical system (BioMEMS) whose operating behavior relies on surface acoustic wave (SAW) manipulated fluid flow in a microchannel. The BioMEMS consists of a separation channel with three inflow channels for injection of the carrier fluid and the cells, two outflow channels for separation, and an interdigital transducer (IDT) close to the lateral wall of the separation channel for generation of the SAWs. The cells can be distinguished by fluorescence. The inflow velocities are tuned so that without SAW actuation a cell of type I leaves the device through a designated outflow channel. However, if a cell of type II is detected, the IDT is switched on and the SAWs modify the fluid flow so that the cell leaves the separation channel through the other outflow boundary. The motion of a cell in the carrier fluid is modeled by the Finite Element Immersed Boundary method (FE-IB). Numerical results are presented that illustrate the feasibility of the surface acoustic wave actuated cell sorting approach.

Adaptive Path Following Primal Dual Interior Point Methods for Shape Optimization of Linear and Nonlinear Stokes Flow Problems

We are concerned with structural optimization problems in CFD where the state variables are supposed to satisfy a linear or nonlinear Stokes system and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. The discretization is taken care of by Taylor-Hood elements with respect to a simplicial triangulation of the computational domain. The efficient numerical solution of the discretized problem relies on adaptive path-following techniques featuring a predictor-corrector scheme with inexact Newton solves of the KKT system by means of an iterative null-space approach. The performance of the suggested method is documented by several illustrative numerical examples.

Path-following primal-dual interior-point methods for shape optimization of stationary flow problems

We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a Bézier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor–Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples.

Adaptive Multilevel Interior-Point Methods in PDE Constrained Optimization

We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. As an application in life sciences, we consider the optimal shape design of capillary barriers in microfluidic biochips.

The Finite Element Immersed Boundary Method for the Numerical Simulation of the Motion of Red Blood Cells in Microfluidic Flows

We study the mathematical modeling and numerical simulation of the motion of red blood cells (RBCs) subject to an external incompressible flow in a microchannel. RBCs are viscoelastic bodies consisting of a deformable elastic membrane enclosing an incompressible fluid. We study two versions of the Finite Element Immersed Boundary Method (FE-IB), a semi-explicit scheme that requires a CFL-type stability condition and a fully implicit scheme that is unconditionally stable and numerically realized by a predictor-corrector continuation strategy featuring an adaptive choice of the time step sizes. The performance of the two schemes is illustrated by numerical simulations for various scenarios including the tank treading motion in microchannels and the motion through thin capillaries.

Projection based model reduction for optimal design of the time-dependent stokes system

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In particular, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. A convenient methodology to realize this idea consists in a suitable combination of domain decomposition techniques and balanced truncation model reduction. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modeling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.

High order approximations in space and time of a sixth order Cahn–Hilliard equation

Abstract We consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C0 conforming Lagrangian finite elements of polynomial degree k ≥ 2. This leads to an initial value problem for an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order p ≥ 2. The resulting parameter dependent nonlinear algebraic system is solved numerically by a predictor-corrector continuation strategy with constant continuation as a predictor and New- ton’s method as a corrector featuring an adaptive choice of the continuation parameter. Numerical results illustrate the performance of the suggested approach.

On the convergence of right transforming iterations for the numerical solution of PDE‐constrained optimization problems

SUMMARY We present an iterative solver, called right transforming iterations (or right transformations), for linear systems with a certain structure in the system matrix, such as they typically arise in the framework of Karush–Kuhn–Tucker (KKT) conditions for optimization problems under PDE constraints. The construction of the right transforming scheme depends on an inner approximate solver for the underlying PDE subproblems. We give a rigorous convergence proof for the right transforming iterative scheme in dependence on the convergence properties of the inner solver. Provided that a fast subsolver is available, this iterative scheme represents an efficient way of solving first-order optimality conditions. Numerical examples endorse the theoretically predicted contraction rates. Copyright

C\(^0\)-Interior Penalty Discontinuous Galerkin Approximation of a Sixth-Order Cahn-Hilliard Equation Modeling Microemulsification Processes

Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C\(^0\) Interior Penalty Discontinuous Galerkin (C\(^0\)IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C\(^0\) conforming Lagrangian finite elements of polynomial degree \(p \ge 2\). For the semidiscretized problem we derive quasi-optimal a priori error estimates for the global discretization error in a mesh-dependent C\(^0\)IPDG norm. The semidiscretized problem represents an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order \( q \ge 2\). Numerical results show the formation of microemulsions in an oil/water system and confirm the theoretically derived convergence rates.