Recent Advances in Computational Engineering

The spray cleaning of surfaces is a standard task in the food and pharmaceutical industries. At present, the development of such nozzles is based on semi-empirical methods, experience and iterative prototyping. This almost makes it prohibitive to develop nozzles for specific customer requirements due to higher time and cost. A Virtual Engineering approach to design and optimize unlimited number of nozzle designs can overcome this.

In this work, a parametric study is carried out to recognize design parameters that have maximum impact on flow. Based on this, a Multiobjective optimization code based on Genetic Algorithm is developed to optimize the design parameters of a full cone nozzle. CFD Simulations were used to estimate the objective functions.

In future, the work shall be extended by comparing genetic algorithm with other optimization algorithms and replacing expensive CFD simulations with meta-models.

We consider the optimal control of fluid-structure interaction (FSI) problems. In order to apply a multilevel optimization algorithm, we introduce a reduced order model using proper orthogonal decomposition (POD). The stability of the resulting system and the existence and uniqueness of solutions is ensured by enriching the reduced basis by suitable supremizers. Further, we reduce the coupling condition by reinterpretation of the variables as boundary integrals. Based on this reduced order model we describe a multilevel optimization algorithm which solves optimal control problems using a sequential quadratic programming (SQP) algorithm and show results for a benchmark problem.

Within the framework of this study, two hybrid LES/RANS turbulence models, namely the limited numerical scales model (LNS) and the very large eddy simulation model (VLES), are investigated for the aeroacoustic simulation based on a NACA0012 test case. The angle of attack is set to 0^∘, 10.8^∘ and 14.4^∘. The simulation results are compared with that of the well-established large eddy simulation (LES) model.

Coping with the so called high Weissenberg number problem (HWNP) is a key focus of research in computational rheology. By numerically simulating viscoelastic flow a breakdown in convergence often occurs for different computational approaches at critically high values of the Weissenberg number. This is due to two major problems concerning stability in the discretization. First, we have a mixed hyperbolic-elliptic problem weighted by a ratio parameter between retardation and relaxation time of viscoelastic fluid. Second, we have a convection-dominated convection-diffusion problem in the constitutive equations. We introduce a solver for viscoelastic Oldroyd B flow with an exclusively high-order Discontinuous Galerkin (DG) scheme for all equations using a local DG formulation in order to solve the hyperbolic constitutive equations and using a streamline upwinding formulation for the convective fluxes of the constitutive equations. The successful implementation of the local DG formulation for Newtonian fluid with appropriate fluxes containing stabilizing penalty parameters is shown in two results. First, a h ^k-convergence study is presented for a non-polynomial manufactured solution for the Stokes system. Second, numerical results are shown for the confined cylinder benchmark problem for Navier-Stokes flow and compared to the same flow using a symmetric interior penalty method without additional constitutive equations.

We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Loève expansion. Regarding iteration numbers, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

Many simulations in Computational Engineering suffer from slow convergence rates of their linear solvers. This is also true for the Finite Pointset Method (FPM), which is a Meshfree Method used in Computational Fluid Dynamics. FPM uses Generalized Finite Difference Methods (GFDM) in order to discretize the arising differential operators. Like other Meshfree Methods, it does not involve a fixed mesh; FPM uses a point cloud instead. We look at the properties of linear systems arising from GFDM on point clouds and their implications on different types of linear solvers, specifically focusing on the differences between one-level solvers and Multigrid Methods, including Algebraic Multigrid (AMG). With the knowledge about the properties of the systems, we develop a new Multigrid Method based on point cloud coarsening. Numerical experiments show that our Multicloud method has the same advantages as other Multigrid Methods; in particular its convergence rate does not deteriorate when refining the point cloud. In future research, we will examine its applicability to a broader range of problems and investigate its advantages in terms of computational performance.

The trend of future massively parallel computer architectures challenges the exploration of additional degrees of parallelism also in the time dimension when solving continuum mechanical partial differential equations. The Adomian decomposition method (ADM) is investigated to this respect in the present work. This is accomplished by comparison with the Runge-Kutta (RK) time integration and put in the context of the viscous Burgers equation.

Our studies show that both methods have similar restrictions regarding their maximal time step size. Increasing the order of the schemes leads to larger errors for the ADM compared to RK. However, we also discuss a parallelization within the ADM, reducing its runtime complexity from O(n ²) to O(n). This indicates the possibility to make it a viable competitor to RK, as fewer function evaluations have to be done in serial, if a high order method is desired. Additionally, creating ADM schemes of high-order is less complex than it is with RK.

Melting water-ice systems develop complex spatio-temporal interface dynamics and a non-trivial temperature field. In this contribution, we present computational aspects of a recently conducted validation study that aims at investigating the role of natural convection for cryo-interface dynamics of water-ice. We will present an established fixed grid model known as the enthalpy porosity method (Brent et al., Numer Heat Transf A 13(3):297–318, 1988; Kumar and Krishna, Energy Procedia 109:314–321, 2017). It is based on introducing a phase field and employs mixture theory. The resulting PDEs are solved using a finite volume discretization. The second part is devoted to experiments that have been conducted for model validation. The evolving water-ice interface is tracked based on optical images that show both the water and the ice phase. To segment the phases, we use a binary Mumford Shah method, which yields a piece-wise constant approximation of the imaging data. Its jump set is the reconstruction of the measured phase interface. Our combined simulation and segmentation effort finally enables us to compare the modeled and measured phase interfaces continuously. We conclude with a discussion of our findings.

This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate according to a probability measure in the set-up of the reduced space. The error of stochastic finite element solutions is usually measured in a root mean square sense regarding their dependence on the stochastic input parameters. An orthogonal projection of a snapshot set onto a corresponding POD basis defines an optimum reduced approximation in terms of a Monte Carlo discretization of the root mean square error. The errors of a weighted POD-greedy Galerkin solution are compared against an orthogonal projection of the underlying snapshots onto a POD basis for a numerical example involving thermal conduction. In particular, it is assessed whether a weighted POD-greedy solutions is able to come significantly closer to the optimum than a non-weighted equivalent. Additionally, the performance of a weighted POD-greedy Galerkin solution is considered with respect to the mean absolute error of an adjoint-corrected functional of the reduced solution.

Thermoforming is a process for the cheap mass-production of thin-walled plastic parts. A sheet of plastic is heated for increased deformability, and then deformed by overpressure into a mold with the end product’s shape. The main drawback is the inhomogeneous wall thickness distribution resulting from the common process. The authors believe that it is possible to improve these inhomogeneities by locally influencing the highly temperature-dependent material strength using directed jets of pressurized air for local cooling. As the high number of potentially influential parameters renders purely experimental parameter studies infeasible, a computational model that couples the flow of the pressurized air with the structural simulation of the deforming plastic is set up. With the combined results of experiments and simulations, a parameter study can be conducted, which allows for an optimization of air flow parameters for a more evenly distributed wall thickness.

Phase interfaces in melting and solidification processes are strongly affected by the presence of convection in the liquid. One way of modeling their transient evolution is to couple an incompressible flow model to an energy balance in enthalpy formulation. Two strong nonlinearities arise, which account for the viscosity variation between phases and the latent heat of fusion at the phase interface.

The resulting coupled system of PDE’s can be solved by a single-domain semi-phase-field, variable viscosity, finite element method with monolithic system coupling and global Newton linearization (Danaila et al., J Comput Phys 274:826–840, 2014). A robust computational model for realistic phase-change regimes furthermore requires a flexible implementation based on sophisticated mesh adaptivity. In this article, we present first steps towards implementing such a computational model into a simulation tool which we call Phaseflow (Zimmerman, https://​github.​com/​geo-fluid-dynamics/​phaseflow-fenics).

Phaseflow utilizes the finite element software FEniCS (Alnæs et al., Arch. Numer Softw 3(100):9–23, 2015), which includes a dual-weighted residual method for goal-oriented adaptive mesh refinement. Phaseflow is an open-source, dimension-independent implementation that, upon an appropriate parameter choice, reduces to classical benchmark situations including the lid-driven cavity and the Stefan problem. We present and discuss numerical results for these, an octadecane PCM convection-coupled melting benchmark, and a preliminary 3D convection-coupled melting example, demonstrating the flexible implementation. Though being preliminary, the latter is, to our knowledge, the first published 3D result for this method. In our work, we especially emphasize reproducibility and provide an easy-to-use portable software container using Docker (Boettiger, ACM SIGOPS Oper Syst Rev 49(1):71–79, 2015).