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Timon Rabczuk

Publications and source records attributed to Timon Rabczuk.

At least 55 records · Page 3Linked to original sources
arXiv2024

Variational Physics-informed Neural Operator (VINO) for Solving Partial Differential Equations

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO's superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.

math.AP↗
arXiv2024

DCEM: A deep complementary energy method for solid mechanics

In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, Physics-Informed Neural Networks (PINNs) and the Deep Energy Method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well-known Deep Energy Method (DEM) is based on the principle of minimum potential energy, but there lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM-Plus (DCEM-P), adding terms that satisfy partial differential equations. Furthermore, we propose a deep complementary energy operator method (DCEM-O) by combining operator learning with physical equations. Initially, we train DCEM-O using high-fidelity numerical results and then incorporate complementary energy. DCEM-P and DCEM-O further enhance the accuracy and efficiency of DCEM.

cs.LG↗
arXiv2024

Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

cs.LG↗
arXiv2024

A discontinuous Galerkin method based isogeometric analysis framework for flexoelectricity in micro-architected dielectric solids

Flexoelectricity - the generation of electric field in response to a strain gradient - is a universal electromechanical coupling, dominant only at small scales due to its requirement of high strain gradients. This phenomenon is governed by a set of coupled fourth-order partial differential equations (PDEs), which require $C^1$ continuity of the basis in finite element methods for the numerical solution. While Isogeometric analysis (IGA) has been proven to meet this continuity requirement due to its higher-order B-spline basis functions, it is limited to simple geometries that can be discretized with a single IGA patch. For the domains, e.g., architected materials, requiring more than one patch for discretization IGA faces the challenge of $C^0$ continuity across the patch boundaries. Here we present a discontinuous Galerkin method-based isogeometric analysis framework, capable of solving fourth-order PDEs of flexoelectricity in the domain of truss-based architected materials. An interior penalty-based stabilization is implemented to ensure the stability of the solution. The present formulation is advantageous over the analogous finite element methods since it only requires the computation of interior boundary contributions on the boundaries of patches. As each strut can be modeled with only two trapezoid patches, the number of $C^0$ continuous boundaries is largely reduced. Further, we consider four unique unit cells to construct the truss lattices and analyze their flexoelectric response. The truss lattices show a higher magnitude of flexoelectricity compared to the solid beam, as well as retain this superior electromechanical response with the increasing size of the structure. These results indicate the potential of architected materials to scale up the flexoelectricity to larger scales, towards achieving universal electromechanical response in meso/macro scale dielectric materials.

cs.CE↗
arXiv2023

CAD-compatible structural shape optimization with a movable Bézier tetrahedral mesh

This paper presents the development of a complete CAD-compatible framework for structural shape optimization in 3D. The boundaries of the domain are described using NURBS while the interior is discretized with Bézier tetrahedra. The tetrahedral mesh is obtained from the mesh generator software Gmsh. A methodology to reconstruct the NURBS surfaces from the triangular faces of the boundary mesh is presented. The description of the boundary is used for the computation of the analytical sensitivities with respect to the control points employed in surface design. Further, the mesh is updated at each iteration of the structural optimization process by a pseudo-elastic moving mesh method. In this procedure, the existing mesh is deformed to match the updated surface and therefore reduces the need for remeshing. Numerical examples are presented to test the performance of the proposed method. The use of the movable mesh technique results in a considerable decrease in the computational effort for the numerical examples.

cs.CE↗
arXiv2023

Boosting output performance of contact-separation mode triboelectric nanogenerators by adopting discontinuity and fringing effect: experiment and modelling studies

Triboelectric nanogenerators (TENGs) are promising self-powering supplies for a diverse range of intelligent sensing and monitoring devices, especially due to their capability of harvesting electric energy from low frequency and small-scale mechanical motions. Inspired by the fact that contact-separation mode TENGs with small contact areas harvest high electrical outputs due to fringing effect, this study employed discontinuity on the dielectric side of contact-separation mode TENGs to promote fringing electric fields for the enhancement of electrical outputs. The results reveal that the TENGs with more discontinuities show higher overall electric performance. Compared to pristine TENGs, the TENGs with cross discontinuities increased the surface charge by 50% and the power density by 114%. However, one should avoid generating discontinuities on tribonegative side of TENGs using metal blade within a positive-ion atmosphere due to the neutralization through electrically conductive metal blade. The computational simulation validated that the TENGs with discontinuities obtained higher electrical outputs, and further investigated the effect of discontinuity gap size and array distance on TENGs performance. This study has provided a promising method for the future design of TENGs using discontinuous structures.

physics.app-ph↗
arXiv2023

A Parallel Feature-preserving Mesh Variable Offsetting Method with Dynamic Programming

Mesh offsetting plays an important role in discrete geometric processing. In this paper, we propose a parallel feature-preserving mesh offsetting framework with variable distance. Different from the traditional method based on distance and normal vector, a new calculation of offset position is proposed by using dynamic programming and quadratic programming, and the sharp feature can be preserved after offsetting. Instead of distance implicit field, a spatial coverage region represented by polyhedral for computing offsets is proposed. Our method can generate an offsetting model with smaller mesh size, and also can achieve high quality without gaps, holes, and self-intersections. Moreover, several acceleration techniques are proposed for the efficient mesh offsetting, such as the parallel computing with grid, AABB tree and rays computing. In order to show the efficiency and robustness of the proposed framework, we have tested our method on the quadmesh dataset, which is available at [https://www.quadmesh.cloud]. The source code of the proposed algorithm is available on GitHub at [https://github.com/iGame-Lab/PFPOffset].

cs.GR↗
arXiv2023

Dual-support smoothed particle hydrodynamics in solid: variational principle and implicit formulation

In this paper, we derive the dual-support smoothed particle hydrodynamics (DS-SPH) in solid in the framework of variational principle. The tangent stiffness matrix of SPH is obtained with ease, which can be served as the basis for implicit SPH. We propose a hourglass energy functional, which allows the direct derivation of hourglass force and hourglass tangent stiffness matrix. The dual-support is identified in all derivation based on variational principles and is automatically satisfied in the assembling of stiffness matrix. The implementation of stiffness matrix comprises with two steps, the nodal assembly based on deformation gradient and global assembly. Several numerical examples are presented to validate the method.

physics.comp-ph↗
arXiv2023

Hexagonal boron-carbon fullerene heterostructures; Stable two-dimensional semiconductors with remarkable stiffness, low thermal conductivity and flat bands

Among exciting recent advances in the field of two-dimensional (2D) materials, the successful fabrications of the C60 fullerene networks has been a particularly inspiring accomplishment. Motivated by the recent achievements, herein we explore the stability and physical properties of novel hexagonal boron-carbon fullerene 2D heterostructures, on the basis of already synthesized B40 and C36 fullerenes. By performing extensive structural minimizations of diverse atomic configurations using the density functional theory method, for the first time, we could successfully detect thermally and dynamically stable boron-carbon fullerene 2D heterostructures. Density functional theory results confirm that the herein predicted 2D networks exhibit very identical semiconducting electronic natures with topological flat bands. Using the machine learning interatomic potentials, we also investigated the mechanical and thermal transport properties. Despite of different bonding architectures, the room temperature lattice thermal conductivity of the predicted nanoporous fullerene heterostructures was found to range between 4 to 10 W/mK. Boron-carbon fullerene heterostructures are predicted to show anisotropic but also remarkable mechanical properties, with tensile strengths and elastic modulus over 8 and 70 GPa, respectively. This study introduces the possibility of developing a novel class of 2D heterostructures based on the fullerene cages, with attractive electronic, thermal and mechanical features.

cond-mat.mtrl-sci↗
arXiv2023

Multi-scale modeling in thermal conductivity of Polyurethane incorporated with Phase Change Materials using Physics-Informed Neural Networks

Polyurethane (PU) possesses excellent thermal properties, making it an ideal material for thermal insulation. Incorporating Phase Change Materials (PCMs) capsules into Polyurethane (PU) has proven to be an effective strategy for enhancing building envelopes. This innovative design substantially enhances indoor thermal stability and minimizes fluctuations in indoor air temperature. To investigate the thermal conductivity of the PU-PCM foam composite, we propose a hierarchical multi-scale model utilizing Physics-Informed Neural Networks (PINNs). This model allows accurate prediction and analysis of the material's thermal conductivity at both the meso-scale and macro-scale. By leveraging the integration of physics-based knowledge and data-driven learning offered by PINNs, we effectively tackle inverse problems and address complex multi-scale phenomena. Furthermore, the obtained thermal conductivity data facilitates the optimization of material design. To fully consider the occupants' thermal comfort within a building envelope, we conduct a case study evaluating the performance of this optimized material in a single room. Simultaneously, we predict the energy consumption associated with this scenario. All outcomes demonstrate the promising nature of this design, enabling passive building energy design and significantly improving occupants' comfort. The successful development of this PINNs-based multi-scale model holds immense potential for advancing our understanding of PU-PCM's thermal properties. It can contribute to the design and optimization of materials for various practical applications, including thermal energy storage systems and insulation design in advanced building envelopes.

physics.comp-ph↗
arXiv2023

Isogeometric analysis of insoluble surfactant spreading on a thin film

In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured.

math.NA↗
arXiv2023

Bond-based nonlocal models by nonlocal operator method in symmetric support domain

This paper is concerned with the energy decomposition of various nonlocal models, including elasticity, thin plates, and gradient elasticity, to arrive at bond-based nonlocal models in which the bond force depends only on the deformation of a single bond. By assuming an appropriate form of bond force and using energy equivalence between local and nonlocal models, several very concise bond-based models are derived. We also revisit the nonlocal operator methods and study the simplified form of second-order NOM in the symmetric support domain. A bent-bond consisting of three points is proposed to describe the curvature and moment. To model the damage, a rule based on Griffith theory for the critical normal strain of the bond is proposed in analogy to the phase field model, which can be applied individually to each bond and provides strain localization. With this rule, the crack direction can be automatically predicted by simply cutting the bond, giving comparable results to the phase field method. At the same time, a damage rule for critical shear strains in shear fractures is proposed. Furthermore, an incremental form of the plasticity model for bond reaction force is derived. Several numerical examples are presented to further validate the nonlocal bond-based models.

physics.class-ph↗
arXiv2023

Phase-field modeling of fluid-driven dynamic cracking in porous media

A phase field model for fluid-driven dynamic crack propagation in poroelastic media is proposed. Therefore, classical Biot poroelasticity theory is applied in the porous medium while arbitrary crack growth is naturally captured by the phase field model. We also account for the transition of the fluid property from the intact medium to the fully broken one by employing indicator functions. We employ a staggered scheme and implement our approach into the software package COMSOL Multiphysics. Our approach is first verified through three classical benchmark problems which are compared to analytical solutions for dynamic consolidation and pressure distribution in a single crack and in a specimen with two sets of joints. Subsequently, we present several 2D and 3D examples of dynamic crack branching and their interaction with pre-existing natural fractures. All presented examples demonstrate the capability of the proposed approach of handling dynamic crack propagation, branching and coalescence of fluid-driven fracture.

math.NA↗
arXiv2023

Phase field modeling of brittle compressive-shear fractures in rock-like materials: A new driving force and a hybrid formulation

Compressive-shear fracture is commonly observed in rock-like materials. However, this fracture type cannot be captured by current phase field models (PFMs), which have been proven an effective tool for modeling fracture initiation, propagation, coalescence, and branching in solids. The existing PFMs also cannot describe the influence of cohesion and internal friction angle on load-displacement curve during compression tests. Therefore, to develop a new phase field model that can simulate well compressive-shear fractures in rock-like materials, we construct a new driving force in the evolution equation of phase field. Strain spectral decomposition is applied and only the compressive part of the strain is used in the new driving force with consideration of the influence of cohesion and internal friction angle. For ease of implementation, a hybrid formulation is established for the phase field modeling. Then, we test the brittle compressive-shear fractures in uniaxial compression tests on intact rock-like specimens as well as those with a single or two parallel inclined flaws. All numerical results are in good agreement with the experimental observation, validating the feasibility and practicability of the proposed PFM for simulating brittle compressive-shear fractures.

cs.CE↗
arXiv2023

Phase field method for quasi-static hydro-fracture in porous media under stress boundary condition considering the effect of initial stress field

Phase field model (PFM) is an efficient fracture modeling method and has high potential for hydraulic fracturing (HF). However, the current PFMs in HF do not consider well the effect of in-situ stress field and the numerical examples of porous media with stress boundary conditions were rarely presented. The main reason is that if the remote stress is applied on the boundaries of the calculation domain, there will be relatively large deformation induced on these stress boundaries, which is not consistent with the engineering observations. To eliminate this limitation, this paper proposes a new phase field method to describe quasi-static hydraulic fracture propagation in porous media subjected to stress boundary conditions, and the new method is more in line with engineering practice. A new energy functional, which considers the effect of initial in-situ stress field, is established and then it is used to achieve the governing equations for the displacement and phase fields through the variational approach. Biot poroelasticity theory is used to couple the fluid pressure field and the displacement field while the phase field is used for determining the fluid properties from the intact domain to the fully broken domain. In addition, we present several 2D and 3D examples to show the effects of in-situ stress on hydraulic fracture propagation. The numerical examples indicate that under stress boundary condition our approach obtains correct displacement distribution and it is capable of capturing complex hydraulic fracture growth patterns.

cs.CE↗
arXiv2023

Ductile fracture modeling by phase field, Hencky strain elasticity and finite J2 plasticity using nonlocal operator method

A phase field model for ductile fracture considering Hencky strain and finite J2 plasticity is presented using the nonlocal operator method. A variational derivation of J2 plasticity at finite strain with a phase field model is performed. The method includes a logarithmic strain tensor and an exponential mapping in the plasticity evolution. A spectral decomposition based algorithm for computing the first and second order derivatives of the composite matrix function is implemented. A consistent tangential stiffness matrix is derived and used in Newton-Raphson iterations. Several numerical examples are performed to validate the method, including notched single-edged plates with brittle fracture or ductile fracture and necking of a bar with/without phase field model.

cond-mat.mtrl-sci↗
arXiv2022

Exploration of mechanical, thermal conductivity and electromechanical properties of graphene nanoribbon springs

Recent experimental advances [Liu \textit{et al., npj 2D Materials and Applications}, 2019, \textbf{3}, 23] propose the design of graphene nanoribbon spring (GNRS) to substantially enhance the stretchability of pristine graphene. GNRS is a periodic undulating graphene nanoribbon, where undulations are of sinus or half-circles or horseshoe shapes. Besides those, GNRS geometry depends on design parameters, like pitch's length and amplitude, thickness and joining angle. Because of the fact that parametric influence on the resulting physical properties are expensive and complicated to be examined experimentally, we explore the mechanical, thermal and electromechanical properties of GNRS using molecular dynamics simulations. Our results demonstrate that horseshoe shape design of GNRS (GNRH) can distinctly outperform the graphene kirigami design concerning the stretchability. The thermal conductivity of GNRS were also examined by developing a multiscale modeling, which suggests that the thermal transport along these nanostructures can be effectively tuned. We found that however, the tensile stretching of GNRS and GNRH does not yield any piezoelectric polarization. The bending induced hybridization change results in a flexoelectric polarization, where the corresponding flexoelectric coefficient is $25\%$ higher than graphene. Our results provide a comprehensive vision to the critical physical properties of GNRS and may help to employ the outstanding physics of the graphene to design novel stretchable nanodevices.

cond-mat.mtrl-sci↗
arXiv2022

Variational energy based XPINNs for phase field analysis in brittle fracture

Modeling fracture is computationally expensive even in computational simulations of two-dimensional problems. Hence, scaling up the available approaches to be directly applied to large components or systems crucial for real applications become challenging. In this work. we propose domain decomposition framework for the variational physics-informed neural networks to accurately approximate the crack path defined using the phase field approach. We show that coupling domain decomposition and adaptive refinement schemes permits to focus the numerical effort where it is most needed: around the zones where crack propagates. No a priori knowledge of the damage pattern is required. The ability to use numerous deep or shallow neural networks in the smaller subdomains gives the proposed method the ability to be parallelized. Additionally, the framework is integrated with adaptive non-linear activation functions which enhance the learning ability of the networks, and results in faster convergence. The efficiency of the proposed approach is demonstrated numerically with three examples relevant to engineering fracture mechanics. Upon the acceptance of the manuscript, all the codes associated with the manuscript will be made available on Github.

cs.CE↗