Thomas Wick

Timo Heister, Thomas Wick

We present a scalable, parallel implementation of a solver for the solution of a phase-field model for quasi-static brittle fracture. The code is available as open source. Numerical solutions in 2d and 3d with adaptive mesh refinement show optimal scaling of the linear solver based on algebraic multigrid, and convergence of the phase-field model towards exact values of functionals of interests such as the crack opening displacement or the total crack volume. In contrast to uniform refinement, adaptive mesh refinement allows us to recover optimal convergence rates for the non-smooth solutions encountered in typical test problems. We also present numerical studies of the influence of the finite domain size on functional evaluations used to approximate the infinite domain.

Sanghyun Lee, Thomas Wick, Mary F. Wheeler

In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.

Nima Noii, Fadi Aldakheel, Thomas Wick et al.

This work addresses an efficient Global-Local approach supplemented with predictor-corrector adaptivity applied to anisotropic phase-field brittle fracture. The phase-field formulation is used to resolve the sharp crack surface topology on the anisotropic/non-uniform local state in the regularized concept. To resolve the crack phase-field by a given single preferred direction, second-order structural tensors are imposed to both the bulk and crack surface density functions. Accordingly, a split in tension and compression modes in anisotropic materials is considered. A Global-Local formulation is proposed, in which the full displacement/phase-field problem is solved on a lower (local) scale, while dealing with a purely linear elastic problem on an upper (global) scale. Robin-type boundary conditions are introduced to relax the stiff local response at the global scale and enhancing its stabilization. Another important aspect of this contribution is the development of an adaptive Global-Local approach, where a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different finite element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. Several numerical results substantiate our developments.

Stefan Frei, Thomas Richter, Thomas Wick

In this work, we describe a simple finite element approach that is able to resolve weak discontinuities in interface problems accurately. The approach is based on a fixed patch mesh consisting of quadrilaterals, that will stay unchanged independent of the position of the interface. Inside the patches we refine once more, either in eight triangles or in four quadrilaterals, in such a way that the interface is locally resolved. The resulting finite element approach can be considered a fitted finite element approach. In our practical implementation, we do not construct this fitted mesh, however. Instead, the local degrees of freedom are included in a parametric way in the finite element space, or to be more precise in the local mappings between a reference patch and the physical patches. We describe the implementation in the open source C++ finite element library deal.II in detail and present two numerical examples to illustrate the performance of the approach. Finally, detailed studies of the behavior of iterative linear solvers complement this work.

Mats Kirkesæther Brun, Thomas Wick, Inga Berre et al.

This paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi-static brittle fracture propagation models, where the fracture evolution is tracked by a phase field variable. The model we consider is a two-field variational inequality system, with the phase field function and the elastic displacements of the solid material as independent variables. Using a penalization strategy, this variational inequality system is transformed into a variational equality system, which is the formulation we take as the starting point for our algorithmic developments. The proposed scheme involves a partitioning of this model into two subproblems; phase field and mechanics, with added stabilization terms to both subproblems for improved efficiency and robustness. We analyze the convergence of the proposed scheme using a fixed point argument, and find that under a natural condition, the elastic mechanical energy remains bounded, and, if the diffusive zone around crack surfaces is sufficiently thick, monotonic convergence is achieved. Finally, the proposed scheme is validated numerically with several bench-mark problems.

Daniel Jodlbauer, Ulrich Langer, Thomas Wick

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix based approach within the Finite Element Method requires lots of memory, which eventually becomes a serious bottleneck. A matrix-free approach overcomes this problems and greatly reduces the amount of required memory, allowing to solve larger problems on available hardware. One key challenge is concerned with the crack irreversibility for which a primal-dual active set method is employed. Here, the Active-Set values of fine meshes must be available on coarser levels of the multigrid algorithm. The developed multigrid method provides a preconditioner for a generalized minimal residual solver. This method is used for solving the linear equations inside Newton's method for treating the overall nonlinear-monolithic discrete displacement/phase-field formulation. Several numerical examples demonstrate the performance and robustness of our solution technology. Mesh refinement studies, variations in the phase-field regularization parameter, iterations numbers of the linear and nonlinear solvers, and some parallel performances are conducted to substantiate the efficiency of the proposed solver for single fractures, multiple pressurized fractures, and a L-shaped panel test in three dimensions.

Bernhard Endtmayer, Ulrich Langer, Ira Neitzel et al.

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples.

Bernhard Endtmayer, Ulrich Langer, Thomas Wick

In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests.

Saurabh Saini, Kapil Ahuja, Thomas Wick et al.

Optimization is essential in deep learning. The foundational method upon which most optimizers are built is momentum-based stochastic gradient descent. However, it suffers from two key drawbacks. First, it has noisy and varying gradients, and second, it has an overshoot phenomenon. To address noisy gradients, Adam was proposed, which remains the most widely used adaptive optimizer. To address the overshoot phenomenon, a control-theory-based PID optimizer was proposed. To tackle both the limitations within a single framework, several variants of Adaptive PID (AdaPID) have recently been proposed. Although AdaPID performs well, it still inherits two critical drawbacks from Adam, namely convergence and stability issues. In this work, we address both these limitations. To fix the convergence issue, we uniquely integrate the idea of using a non-increasing effective learning rate into AdaPID (originally proposed in AMSGrad, an extension of Adam). To fix the stability issue, we innovatively integrate a gradient difference based modulation factor into AdaPID (originally proposed in DiffGrad, another extension of Adam). Combining both these ideas in AdaPID, results in our novel IAdaPID-ADG optimizer. We evaluate our proposed optimizer on multiple datasets, including benchmark datasets (MNIST and CIFAR10) and real-world datasets (IARC and AnnoCerv). The IAdaPID-ADG substantially outperforms all competing optimizers. Additionally, we perform an ablation study on the MNIST dataset to demonstrate the contribution of each added component.

Stefan Frei, Tobias Knoke, Marc C. Steinbach et al.

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

Robin Schmöcker, Alexander Henkes, Julian Roth et al.

This works investigates the generalization capabilities of MeshGraphNets (MGN) [Pfaff et al. Learning Mesh-Based Simulation with Graph Networks. ICML 2021] to unseen geometries for fluid dynamics, e.g. predicting the flow around a new obstacle that was not part of the training data. For this purpose, we create a new benchmark dataset for data-driven computational fluid dynamics (CFD) which extends DeepMind's flow around a cylinder dataset by including different shapes and multiple objects. We then use this new dataset to extend the generalization experiments conducted by DeepMind on MGNs by testing how well an MGN can generalize to different shapes. In our numerical tests, we show that MGNs can sometimes generalize well to various shapes by training on a dataset of one obstacle shape and testing on a dataset of another obstacle shape.

Wenlong He, Thomas Wick, Xiaohe Yue et al.

We study a fluid-poroelasticity interaction (FPSI) problem coupling the unsteady Stokes equations with the fully dynamic Biot system. A major challenge in such problems is to design partitioned schemes that remain robust in locking-related parameter regimes while preserving the physical interface coupling structure.To address this issue, we introduce two auxiliary variables and reformulate the Biot system as a four-field problem consisting of a dynamic Stokes-like system coupled with a diffusion equation. Crucially, this reformulation preserves the original interface conditions. Based on Robin-Robin transmission conditions with explicitly lagged interface data, we construct a fully decoupled scheme in which the fluid and poroelastic subproblems can be solved independently and in parallel at each time step, without sub-iterations.We prove that the resulting method is unconditionally stable and derive optimal-order error estimates in the $H^1$-norm. The analysis further shows that the scheme is robust with respect to extreme poroelastic parameters and avoids the locking effects inherent in standard formulations. Numerical experiments confirm the theoretical convergence results and demonstrate the locking-robust performance of the proposed method.

Saurabh Saini, Kapil Ahuja, Marc C. Steinbach et al.

In this study, we develop a new CAD system for accurate thyroid cancer classification with emphasis on feature extraction. Prior studies have shown that thyroid texture is important for segregating the thyroid ultrasound images into different classes. Based upon our experience with breast cancer classification, we first conjuncture that the Discrete Cosine Transform (DCT) is the best descriptor for capturing textural features. Thyroid ultrasound images are particularly challenging as the gland is surrounded by multiple complex anatomical structures leading to variations in tissue density. Hence, we second conjuncture the importance of localization and propose that the Local DCT (LDCT) descriptor captures the textural features best in this context. Another disadvantage of complex anatomy around the thyroid gland is scattering of ultrasound waves resulting in noisy and unclear textures. Hence, we third conjuncture that one image descriptor is not enough to fully capture the textural features and propose the integration of another popular texture capturing descriptor (Improved Local Binary Pattern, ILBP) with LDCT. ILBP is known to be noise resilient as well. We term our novel descriptor as Binary Pattern Driven Local Discrete Cosine Transform (BPD-LDCT). Final classification is carried out using a non-linear SVM. The proposed CAD system is evaluated on the only two publicly available thyroid cancer datasets, namely TDID and AUITD. The evaluation is conducted in two stages. In Stage I, thyroid nodules are categorized as benign or malignant. In Stage II, the malignant cases are further sub-classified into TI-RADS (4) and TI-RADS (5). For Stage I classification, our proposed model demonstrates exceptional performance of nearly 100% on TDID and 97% on AUITD. In Stage II classification, the proposed model again attains excellent classification of close to 100% on TDID and 99% on AUITD.

Ayan Chakraborty, Thomas Wick, Xiaoying Zhuang et al.

Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work, functions with desired properties are devised to approximate the solutions of PDEs. Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. The novel algorithmic developments are substantiated with numerical test examples. The advantages of using deep neural network over the shallow neural network are demonstrated and the convergence enhancing techniques are also presented

Rohit Agrawal, Kapil Ahuja, Marc C. Steinbach et al.

We hide grayscale secret images into a grayscale cover image, which is considered to be a challenging steganography problem. Our goal is to develop a steganography scheme with enhanced embedding capacity while preserving the visual quality of the stego-image as well as the extracted secret image, and ensuring that the stego-image is resistant to steganographic attacks. The novel embedding rule of our scheme helps to hide secret image sparse coefficients into the oversampled cover image sparse coefficients in a staggered manner. The stego-image is constructed by using ADMM to solve the LASSO formulation of the underlying minimization problem. Finally, the secret images are extracted from the constructed stego-image using the reverse of our embedding rule. Using these components together, to achieve the above mentioned competing goals, forms our most novel contribution. We term our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). We perform extensive experiments on several standard images. By choosing the size of the secret images to be half of the of cover image, we obtain embedding capacities of 2 bpp (bits per pixel), 4 bpp, 6 bpp, and 8 bpp while embedding one, two, three, and four secret images, respectively. Our focus is on hiding multiple secret images. For the case of hiding two and three secret images, our embedding capacities are higher than all the embedding capacities obtained in the literature until now. For the case of hiding four secret images, although our capacity is slightly lower than one work, we do better on the other two goals; a) very little deterioration in the quality of the stego-images and extracted secret images, and b) inherently and designed-to-be resistant to steganographic attacks. Additionally, we demonstrate that SABMIS executes in few minutes, and show its application on two real-life problems.

Mostafa Abbaszadeh, Amirreza Khodadadian, Mehdi Dehghan et al.

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection-reaction-diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank--Nicolson finite difference formulation. In the stochastic direction, we also employ a random variable $W$ based on the $Q-$Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

Vahid Mohammadi, Mehdi Dehghan, Amirreza Khodadadian et al.

In this work, we apply two meshless methods for the numerical solution of the time-dependent transport equation defined on the sphere in spherical coordinates. The first technique, which was introduced by Mirzaei (BIT Numerical Mathematics, 54 (4) 1041-1063, 2017) in Cartesian coordinates is a generalized moving least squares approximation, and the second one, which is developed here, is moving kriging least squares interpolation on the sphere. These methods do not depend on the background mesh or triangulation, and they can be implemented on the transport equation in spherical coordinates easily using different distribution points. Furthermore, the time variable is approximated by a second-order backward differential formula. The obtained fully discrete scheme is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner at each time step. Three well-known test problems namely solid body rotation, vortex roll-up, and deformational flow are solved to demonstrate our developments.

Nima Noii, Thomas Wick

In this work, we extend a phase-field approach for pressurized fractures to non-isothermal settings. Specifically, the pressure and the temperature are given quantities and the emphasis is on the correct modeling of the interface laws between a porous medium and the fracture. The resulting model is augmented with thermodynamical arguments and then analyzed from a mechanical perspective. The numerical solution is based on a robust semi-smooth Newton approach in which the linear equation systems are solved with a generalized minimal residual method and algebraic multigrid preconditioning. The proposed modeling and algorithmic developments are substantiated with different examples in two- and three dimensions. We notice that for some of these configurations manufactured solutions can be constructed, allowing for a careful verification of our implementation. Furthermore, crack-oriented predictor-corrector adaptivity and a parallel implementation are used to keep the computational cost reasonable. Snapshots of iteration numbers show an excellent performance of the nonlinear and linear solution algorithms. Lastly, for some tests, a computational analysis of the effects of strain-energy splitting is performed, which has not been undertaken to date for similar phase-field settings involving pressure, fluids or non-isothermal effects.