O. Ebel, S. Schmidt and A. Walther
Solving non-smooth semi-linear optimal control problems with abs-linearization
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We investigate optimization problems with a non-smooth partial differential equation as constraint, where non-smoothness is assumed to be caused by the functions abs(), min() and max(). For the efficient as well as robust solution of such problems, we propose a new optimization method based on abs-linearisation, i.e., a special handling of the non-smoothness without regularization. The key idea of this approach is the determination of stationary points by an appropriate decomposition of the original non-smooth problem into several smooth so-called branch problems. Each of these branch problems can be solved by classical means. The exploitation of corresponding optimality conditions for the smooth case identifies the next branch and thus yields a successive reduction of the objective value. This approach is able to solve the considered class of non-smooth optimization problems without any regularization of the non-smoothness and additionally maintains reasonable convergence properties. Numerical results for non-smooth optimization problems illustrate the proposed approach and its performance.

K. Kush, S. Schmidt and N. Gauger
An approximate Newton smoothing method for shape optimization
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A novel methodology to efficiently approximate the Hessian for numerical shape optimization is considered. The method enhances operator symbol approximations by including body fitted coordinates and spatially changing symbols in a semi automated framework based on local Fourier analysis. Contrary to classical operator symbol methods, the proposed strategy will identify areas in which a non-smooth design is physically meaningful and will automatically turn off smoothing in these regions. A new strategy to also numerically identify the analytic symbol is derived, extending the procedure to a wide variety of problems. The effectiveness is demonstrated by using drag minimization in Stokes and Navier-Stokes flows.

J. S. Dokken, S. W. Funke, A. Johansson and S. Schmidt
Shape optimization using multimesh FEM with Nitsche coupling
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An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other typically lacks robustness or is computationally expensive. This paper proposes a different approach, in which the computational domain is represented by multiple, independent meshes. A Nitsche based finite element method is used to weakly enforce continuity over the non-matching mesh interfaces. The optimization is preformed using an iterative gradient method, in which the shape-sensitivities are obtained by employing the Hadamard formulas and the adjoint approach. An optimize-then-discretize approach is chosen due to its independence of the FEM framework. Since the individual meshes may be moved freely, re-meshing or mesh deformations can be entirely avoided in cases where the geometry changes consists of rigid motions or scaling. By this free movement, we obtain robust and computational cheap mesh adaptation for optimization problems even for large domain changes. For general geometry changes, the method can be combined with mesh-deformation or re-meshing techniques to reduce the amount of deformation required. We demonstrate the capabilities of the method on several examples, including the optimal placement of heat emitting wires in a cable to minimize the chance of overheating, the drag minimization in Stokes flow, and the orientation of 25 objects in a Stokes flow.

Journal Articles

M. Herrmann, R. Herzog, S. Schmidt and G. Wachsmuth
Discrete Total Variation with finite elements and applications to imaging
Journal of Mathematical Imaging and Vision, 2018 [html , html (preprint) ]
The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TVseminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart–Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L2 and TV-L1, can be implemented in low and higherorder finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.

M. Herrmann, R. Herzog, H. Kröner, S. Schmidt and J. Vidal
Analysis and an Interior Point Approach for TV Image Reconstruction Problems on Smooth Surfaces
SIAM Journal on Imaging Sciences, 11(2):889–922, 2018 [html]
[Lai, Chan (Computer Vision and Image Understanding, 2011)] introduced an analog of the total variation image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. The problem is defined in terms of quantities intrinsic to the surface and it is therefore independent of the parametrization. In this paper, a rigorous analytical framework is developed for this model and its Fenchel predual. It is shown that the predual of the total variation problem is a quadratic optimization problem for the predual vector field p in H(div, S) with pointwise inequality constraints on the surface. As in the flat case, p serves as an edge detector. A function space interior point method is proposed for the predual problem, which is discretized by conforming Raviart-Thomas finite elements on a triangulation of the surface. Well-posedness of the barrier problems is established. Numerical examples including denoising and inpainting problems with both gray-scale and color images on scanned 3D geometries of considerable complexity are presented.

S. Schmidt, M. Schütte, and A. Walther
Efficient numerical solution of geometric inverse problems involving Maxwell’s equations using shape derivatives and automatic code generation
SIAM Journal on Scientific Computing, 40(2):B405–B428, 2018 [html , html (preprint) ]
We propose a novel approach using shape derivatives to solve sharp interface geometric inverse optimization problems governed by Maxwell's equations. A tracking type target functional determines the distance between the solution of a 3D time-dependent Maxwell problem and given measured data in an $L_2$-norm. Minimization is conducted using $H^1$-gradient information based on shape derivatives, which is related to the shape Hessian of the problem regularization. We describe the underlying formulas, the derivation of appropriate upwind fluxes and arrive at shape gradients for general tracking type objectives and conservation laws. Subsequently, an explicit boundary gradient formulation based on variational forms is given for the problem at hand. Using such variational forms as domain specific programming languages, the FEniCS environment can then automatically generate the solvers, leading to structure exploiting data efficient transient adjoints. Checkpointing strategies are not necessary. Numerical results of up to $1.2\cdot 10^9$ state unknowns demonstrate the practicability of the proposed approach.

S. Schmidt
Weak and Strong Form Shape Hessians and Their Automatic Generation
SIAM Journal on Scientific Computing, 40(2):C210–C233, 2018 [html]
By analyzing variational problems formulated in the Unified Form Language, a structure-aware differentiation tool is presented, which can automatically generate both the classical boundary representation and the weak or ''volume'' formulation of first and second order shape derivatives. Where applicable, the tool can either automatically apply the divergence theorem in tangent spaces for the strong form or calculate discrete material derivatives for the weak form. Furthermore, additional assumptions and simplifications can also be automatically applied, such that a repeated application leads to symmetric shape Hessians. The resulting expression can then be processed by the FEniCS environment, resulting in the semi-automatic creation of shape optimization chains from a user-supplied Lagrangian only. The methodology is tested by conducting shape Newton optimization using examples from geometry and CFD. The respective software is released as open source at

S. Schmidt, E. Wadbro, and M. Berggren
Large-scale three-dimensional acoustic horn optimization
SIAM Journal on Scientific Computing, 38(6):B917–B940, 2016 [html , .pdf (preprint) ]
We consider techniques that enable large-scale gradient-based shape optimization of wave-guiding devices in the context of three-dimensional time-domain simulations. The approach relies on a memory efficient boundary representation of the shape gradient together with primal and adjoint solvers semiautomatically generated by the FEniCS framework. The hyperbolic character of the governing linear wave equation, written as a first-order system, is exploited through systematic use of the characteristic decomposition both to define the objective function and to obtain stable numerical fluxes in the discontinuous Galerkin spatial discretization. The methodology is successfully used to optimize the shape of a midrange acoustic horn, described by 1,762 design variables, for maximum transmission efficiency, where the parallel computations involve a total of 3.5*10^9 unknowns.

M. Sonntag, S. Schmidt, and N. Gauger
Shape derivatives for the compressible Navier-Stokes equations in variational form
Journal of Computational and Applied Mathematics, 296:334-351, 2016 [ html | html (preprint) ]
Shape optimization based on surface gradients and the Hadarmard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the "function composition" approach involving local shape derivatives and an alternate methodology based on the weak form of the state equation. The resulting gradient expressions are found to be equal only if the existence of a strong form solution is assumed. Surface shape derivatives based on both formulations are implemented within a Discontinuous Galerkin flow solver of variable order. The gradient expression stemming from the variational approach is found to give superior accuracy when compared to finite differences.

M. Siebenborn, V. Schulz, and S. Schmidt
A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations
Computing and Visualization in Science, 15(2):61-73, 2012 (published online 17.08.2013) [ html | .pdf (preprint) ]
In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.

S. Schmidt, V. Schulz, C. Ilic, and N.R. Gauger.
Three Dimensional Large Scale Aerodynamic Shape Optimization based on Shape Calculus.
AIAA Journal, 51(11):2615-2627, 2013. [ html ]
Large-scale three-dimensional aerodynamic shape optimization based on the compressible Euler equations is considered. Shape calculus is used to derive an exact surface formulation of the gradients, enabling the computation of shape gradient information for each surface mesh node without having to calculate further mesh sensitivities. Special attention is paid to the applicability to large-scale three dimensional problems like the optimization of an Onera M6 wing or a complete blended-wing–body aircraft. The actual optimization is conducted in a one-shot fashion, in which the tangential Laplace operator is used as a Hessian approximation, thereby also preserving the regularity of the shape.

S. Schmidt and V. Schulz.
A 2589 line topology optimization code written for the graphics card.
Computing and Visualization in Science, 14(6):249-256, 2011. [ html | .pdf (preprint) | .pdf (preprint, interactive 3D) ]
We investigate topology optimization based on the solid isotropic material with penalization approach on compute unified device architecture enabled graphics cards in three dimensions. Linear elasticity is solved entirely on the GPU by a matrix-free conjugate gradient method using finite elements. Due to the unique requirements of the single instruction, multiple data stream processor, special attention is given to the procedural generation of matrix–vector prod- ucts entirely on the graphics card. The GPU code is found to be extremely efficient, being faster than a 48 core shared memory CPU system. CPU and GPU implementations show different performance bottlenecks. The sources are available at

S. Schmidt, C. Ilic, V. Schulz, and N.R. Gauger.
Airfoil design for compressible inviscid flow based on shape calculus.
Optimization and Engineering, 12(3):349-369, 2011. [ html  | .pdf ]
Aerodynamic design based on the Hadamard representation of shape gradients is considered. Using this approach, the gradient of an objective function with respect to a deformation of the shape can be computed as a boundary integral without any additional "mesh sensitivities" or volume quantities. The resulting very fast gradient evaluation procedure greatly supports a one-shot optimization strategy and coupled with an appropriate shape Hessian approximation, a very efficient shape opti- mization procedure is created that does not deteriorate with an increase in the number of design parameters. As such, all surface mesh nodes are used as shape design pa- rameters for optimizing a variety of lifting and non-lifting airfoil shapes using the compressible Euler equations to model the fluid.

C. Schillings, S. Schmidt, and V. Schulz.
Efficient Shape Optimization for Certain and Uncertain Aerodynamic Design.
Computers and Fluids, 46(1):78-87, 2011. [ html ]
In this paper, we present novel developments in aerodynamic shape optimization based on shape calculus as well as the proper treatment of aleatoric uncertainties in the field of aerodynamic design.

S. Schmidt and V. Schulz.
Shape derivatives for general objective functions and the incompressible Navier-Stokes equations.
Control and Cybernetics, 39(3):677-713, 2010. [ http ]
The aim of this paper is to present the shape derivative for a wide array of objective functions using the incompressible Navier-Stokes equations as a state constraint. Most real world applications of computational fluid dynamics are shape optimization problems in nature, yet special shape optimization techniques are seldom used outside the field of elliptic partial differential equations and linear elasticity. This article tries to be self contained, also presenting many useful results from the literature. We conclude with a comparison of different objective functions for the shape optimization of an obstacle in a channel, which can be done quite conveniently when one knows the general form of the shape gradient.

K. Eppler, S. Schmidt, V. Schulz, and C. Ilic.
Preconditioning the pressure tracking in fluid dynamics by shape Hessian information.
Journal of Optimization Theory and Applications, 141(3):513-531, 2009. [ DOI | http ]
Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.

S. Schmidt and V. Schulz.
Impulse response approximations of discrete shape Hessians with application in CFD.
SIAM Journal on Control and Optimization, 48(4):2562-2580, 2009. [ DOI | .pdf ]
This paper discusses the symbol of the Hessian of a shape optimization problem in a viscid, incompressible flow. The symbol of the Hessian for the Stokes shape optimization problem is analytically approximated by tracking a Fourier mode through the operators involved. We propose a discrete method for finding the symbol and confirm the correctness by comparison with the analytical data. A preconditioner is constructed for both the Stokes and Navier-Stokes equation, which greatly accelerates the optimization.

S. Schmidt and V. Schulz.
Pareto-curve continuation in multi-objective optimization.
Pacific Journal of Optimization, 4(2):243-257, 2008. [ http ]
Aerodynamic shape optimization usually aims at optimizing a single design objective. This often means a drag or noise reduction given a constant lift or pitching moment. By re-formulating the constraints as additional objective functions, one can embed this scenario into a multi-objective design approach, which results in a set of indifference points between the cost functions. A potential designer can then choose from a variety of equally suitable solutions. In this paper, we explore curve continuation strategies for finding the indifference curve exemplified by an inviscid lift/drag optimization of the RAE2822 airfoil. Special attention is given to approaches that work well with a given SQP solver for the single objective problem. Different parametrizations of the curve are also studied.

Technical Reports

A. Borzi, J. Merger, J. Müller, A. Rosch, C. Schenk, D. Schmidt, S. Schmidt, V. Schulz, K. Velten, C. von Wallbrunn, M. Zänglein
Novel model for wine fermentation including the yeast dying phase
ePrint arXiv:, 1412.6068, 2014 [ http ]
This paper presents a novel model for wine fermentation including a death phase for yeast and the influence of oxygen on the process. A model for the inclusion of the yeast dying phase is derived and compared to a model taken from the literature. The modeling ability of the several models is analyzed by comparing their simulation results.

S. Schmidt
A Two Stage CVT / Eikonal Convection Mesh Deformation Approach for Large Nodal Deformations
ePrint arXiv: 1411.7663, 2014 [ html ]
A two step mesh deformation approach for large nodal deformations, typically arising from non-parametric shape optimization, fluid-structure interaction or computer graphics, is considered. Two major difficulties, collapsed cells and an undesirable parameterization, are overcome by considering a special form of ray tracing paired with a centroid Voronoi reparameterization. The ray direction is computed by solving an Eikonal equation. With respect to the Hadamard form of the shape derivative, both steps are within the kernel of the objective and have no negative impact on the minimizer. The paper concludes with applications in 2D and 3D fluid dynamics and automatic code generation and manages to solve these problems without any remeshing. The methodology is available as a FEniCS shape optimization add-on at

Conference Publications (Selection)

S. Schmidt
Structure exploitation, AD and the continuous problem.
In N. Gauger, M. Giles, M. Gunzburger, and U. Naumann, editors, Adjoint Methods in Computational Science, Engineering, and Finance, volume 4 (9) of Report from Dagstuhl Seminar 14371, pages 23–24. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2015.

S. Schmidt, N. Popescu, M. Berggren, V. Schulz, and A.  Walther.
Large scale shape optimization for deterministic problems.
In M. Heinkenschloss and V. Schulz, editor, Numerical Methods for PDE Constrained Optimization with Uncertain Data, number 4/2013 in Oberwolfach Reports, pages 284-285. Mathematisches Forschungszentrum Oberwolfach, 2013.

S. Schmidt, V. Schulz, C. Ilic, and N.R. Gauger.
Three dimensional large scale aerodynamic shape optimization based on shape calculus.
41st AIAA Fluid Dynamics Conference and Exhibit
AIAA 2011-3718.pdf ]

S. Schmidt, V. Schulz, C. Ilic, and N.R. Gauger.
Large scale aerodynamic shape optimization based on shape calculus.
In I.H. Tuncer, editor, ECCOMAS-CFD&Optimization, 2011-054, ISBN 978-605-61427-4-1, 2011.

S. Schmidt and V. Schulz.
Largescale aerodynamic shape optimization.
In C.M. Elliot, Y. Giga, M. Hinze, and V. Styles, editors, New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries, number 7/2010 in Oberwolfach Reports, pages 307-308. Mathematisches Forschungszentrum Oberwolfach, 2010.
S. Schmidt, C. Ilic, V. Schulz, and N.R. Gauger.
Fast non-parametric large scale aerodynamic shape optimization.
In M. Heinkenschloss, R.H.W. Hoppe, and V. Schulz, editors, Numerical Techniques for Optimization Problems with PDE Constraints, number 4/2009 in Oberwolfach Reports, pages 69-71. Mathematisches Forschungszentrum Oberwolfach, 2009.


S. Schmidt.
Efficient Large Scale Aerodynamic Design Based on Shape Calculus.
PhD. Thesis, University of Trier, 2010html | .pdf ]
S. Schmidt.
Geometric Inverse Problems, Images and SQP Methods Based on Weak Shape Hessians.
Habilitation (cumulative), University of Würzburg, 2018 [.pdf ]