Computer Science

A Trace-Finite-Cell-Method for the numerical analysis of thin shells is presented combining concepts of the TraceFEM and the Finite-Cell-Method. As an underlying shell model we use the Koiter model, which we re-derive in strong form based on first principles of continuum mechanics by recasting well-known relations formulated in local coordinates to a formulation independent of a parametrization. The field approximation is constructed by restricting shape functions defined on a structured background grid on the shell surface. As shape functions we use on a background grid the tensor product of cubic splines. This yields C 1 -continuous approximation spaces, which are required by the governing equations of fourth order. The parametrization-free formulation allows a natural implementation of the proposed method and manufactured solutions on arbitrary geometries for code verification. Thus, the implementation is verified by a convergence analysis where the error is computed with an exact manufactured solution. Furthermore, benchmark tests are investigated.

In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine-scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D-1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi-directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects.

The Earth's climate is rapidly changing and some of the most drastic changes can be seen in the Arctic, where sea ice extent has diminished considerably in recent years. As the Arctic climate continues to change, gathering in situ sea ice measurements is increasingly important for understanding the complex evolution of the Arctic ice pack. To date, observations of ice stresses in the Arctic have been spatially and temporally sparse. We propose a measurement framework that would instrument existing sea ice buoys with strain gauges. This measurement framework uses a Bayesian inference approach to infer ice loads acting on the buoy from a set of strain gauge measurements. To test our framework, strain measurements were collected from an experiment where a buoy was frozen into ice that was subsequently compressed to simulate convergent sea ice conditions. A linear elastic finite element model was used to describe the response of the deformable buoy to mechanical loading, allowing us to link the observed strain on the buoy interior to the applied load on the buoy exterior. The approach presented in this paper presents an instrumentation framework that could use existing buoy platforms as in situ sensors of internal stresses in the ice pack.

The purpose of this work is to train an Encoder-Decoder Convolutional Neural Networks (CNN) to automatically add local fine-scale stress corrections to coarse stress predictions around unresolved microscale features. We investigate to what extent such a framework provides reliable stress predictions inside and outside particular training sets. Incidentally, we aim to develop efficient offline data generation methods to maximise the domain of validity of the CNN predictions. To achieve these ambitious goals, we will deploy a Bayesian approach providing not point estimates, but credible intervals of the fine-scale stress field to evaluate the uncertainty of the predictions. This will automatically encompass the lack of knowledge due to unseen macro and micro features. The uncertainty will be used in a Selective Learning framework to reduce the data requirements of the network. In this work we will investigate stress prediction in 2D porous structures with randomly distributed circular holes.

Bayesian regularization-backpropagation neural network (BR-BPNN) model is employed to predict some aspects of the gecko spatula peeling viz. the variation of the maximum normal and tangential pull-off forces and the resultant force angle at detachment with the peeling angle. K-fold cross validation is used to improve the effectiveness of the model. The input data is taken from finite element (FE) peeling results. The neural network is trained with 75% of the FE dataset. The remaining 25% are utilized to predict the peeling behavior. The training performance is evaluated for every change in the number of hidden layer neurons to determine the optimal network structure. The relative error is calculated to draw a clear comparison between predicted and FE results. It is shown that the BR-BPNN model in conjunction with k-fold technique has significant potential to estimate the peeling behavior.

In this paper an efficient and reliable method for stochastic yield estimation is presented. Since one main challenge of uncertainty quantification is the computational feasibility, we propose a hybrid approach where most of the Monte Carlo sample points are evaluated with a surrogate model, and only a few sample points are reevaluated with the original high fidelity model. Gaussian Process Regression is a non-intrusive method which is used to build the surrogate model. Without many prerequisites, this gives us not only an approximation of the function value, but also an error indicator that we can use to decide whether a sample point should be reevaluated or not. For two benchmark problems, a dielectrical waveguide and a lowpass filter, the proposed methods outperform classic approaches.

This contribution proposes the first 3D beam-to-beam interaction model for molecular interactions between curved slender fibers undergoing large deformations. While the general model is not restricted to a specific beam formulation, in the present work it is combined with the geometrically exact beam theory and discretized via the finite element method. A direct evaluation of the total interaction potential for general 3D bodies requires the integration of contributions from molecule or charge distributions over the volumes of the interaction partners, leading to a 6D integral (two nested 3D integrals) that has to be solved numerically. Here, we propose a novel strategy to formulate reduced section-to-section interaction laws for the resultant interaction potential between a pair of cross-sections of two slender fibers such that only two 1D integrals along the fibers' length directions have to be solved numerically. This section-to-section interaction potential (SSIP) approach yields a significant gain in efficiency, which is essential to enable the simulation of relevant time and length scales for many practical applications. In a first step, the generic structure of SSIP laws, which is suitable for the most general interaction scenario (e.g. fibers with arbitrary cross-section shape and inhomogeneous atomic/charge density within the cross-section) is presented. Assuming circular, homogeneous cross-sections, in a next step, specific analytical expressions for SSIP laws describing short-range volume interactions (e.g. van der Waals or steric interactions) and long-range surface interactions (e.g. Coulomb interactions) are proposed. The validity of the SSIP laws as well as the accuracy and robustness of the general SSIP approach to beam-to-beam interactions is thoroughly verified by means of a set of numerical examples considering steric repulsion, electrostatic or van der Waals adhesion.

A general-purpose computational homogenization framework is proposed for the nonlinear dynamic analysis of membranes exhibiting complex microscale and/or mesoscale heterogeneity characterized by in-plane periodicity that cannot be effectively treated by a conventional method, such as woven fabrics. The framework is a generalization of the "finite element squared" (or FE2) method in which a localized portion of the periodic subscale structure is modeled using finite elements. The numerical solution of displacement driven problems involving this model can be adapted to the context of membranes by a variant of the Klinkel-Govindjee method[1] originally proposed for using finite strain, three-dimensional material models in beam and shell elements. This approach relies on numerical enforcement of the plane stress constraint and is enabled by the principle of frame invariance. Computational tractability is achieved by introducing a regression-based surrogate model informed by a physics-inspired training regimen in which FE 2 is utilized to simulate a variety of numerical experiments including uniaxial, biaxial and shear straining of a material coupon. Several alternative surrogate models are evaluated including an artificial neural network. The framework is demonstrated and validated for a realistic Mars landing application involving supersonic inflation of a parachute canopy made of woven fabric.

Shell analysis is a well-established field, but achieving optimal higher-order convergence rates for such simulations is a difficult challenge. We present an isogeometric Kirchhoff-Love shell framework that treats every numerical aspect in a consistent higher-order accurate way. In particular, a single trimmed B-spline surface provides a sufficiently smooth geometry, and the non-symmetric Nitsche method enforces the boundary conditions. A higher-order accurate reparametrization of cut knot spans in the parameter space provides a robust, higher-order accurate quadrature for (multiple) trimming curves, and the extended B-spline concept controls the conditioning of the resulting system of equations. Besides these components ensuring all requirements for higher-order accuracy, the presented shell formulation is based on tangential differential calculus, and level-set functions define the trimming curves. Numerical experiments confirm that the approach yields higher-order convergence rates, given that the solution is sufficiently smooth.

The Darwin field model addresses an approximation to Maxwell's equations where radiation effects are neglected. It allows to describe general quasistatic electromagnetic field phenomena including inductive, resistive and capacitive effects. A Darwin formulation based on the Darwin-Ampère equation and the implicitly included Darwin-continuity equation yields a non-symmetric and ill-conditioned algebraic systems of equations received from applying a geometric spatial discretization scheme and the implicit backward differentiation time integration method. A two-step solution scheme is presented where the underlying block-Gauss-Seidel method is shown to change the initially chosen gauge condition and the resulting scheme only requires to solve a weakly coupled electro-quasistatic and a magneto-quasistatic discrete field formulation consecutively in each time step. Results of numerical test problems validate the chosen approach.