Unified Variational Framework Research Articles

Phase-field modeling of fracture with physics-informed deep learning

We explore the potential of the deep Ritz method to learn complex fracture processes such as quasistatic crack nucleation, propagation, kinking, branching, and coalescence within the unified variational framework of phase-field modeling of brittle fracture. We elucidate the challenges related to the neural-network-based approximation of the energy landscape, and the ability of an optimization approach to reach the correct energy minimum, and we discuss the choices in the construction and training of the neural network which prove to be critical to accurately and efficiently capture all the relevant fracture phenomena. The developed method is applied to several benchmark problems and the results are shown to be in qualitative and quantitative agreement with the finite element solution. The robustness of the approach is tested by using neural networks with different initializations.

A Unified Variational Framework on Macroscopic Computations for Two-Phase Flow with Moving Contact Lines

A Unified Variational Framework on Macroscopic Computations for Two-Phase Flow with Moving Contact Lines

Multiplicity of nontrivial solutions for p‐Kirchhoff type equation with Neumann boundary conditions

This paper is concerned with the multiplicity results to a class of ‐Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by the abstract linking lemma due to Brézis and Nirenberg. We obtain the twofold results in subcritical and critical cases, which is a meaningful addition and completeness to the known results about Kirchhoff equation. At the same time, this paper also gives a method to deal with ‐Laplacian, Kirchhoff equation, and some Kirchhoff type equation in a unified variational framework.

Overview of phase-field models for fatigue fracture in a unified framework

The phase-field method has gained much attention as a novel method to simulate fracture due to its straightforward way allowing to cover crack initiation and propagation without additional conditions. More recently, it has also been applied to fatigue fracture due to cyclic loading. This publication gives an overview of the main phase-field fatigue models published to date. For the first time, we present all models in a unified variational framework for best comparability. Subsequently, the models are compared regarding their most important features. It becomes apparent that they can be classified in mainly two categories according to the way fatigue is implemented in the model — that is as a gradual degradation of the fracture toughness or as an additional term in the crack driving force. We aim to provide a helpful guide for choosing the appropriate model for different applications and for developing existing models further.

A unified variational framework of no-tension and no-compression solids and its application to finite element analysis

No-tension and no-compression constitutive models have important applications in solid mechanics, such as modelling of masonry, wrinkled membrane, unilateral contact interface, etc. Although lots of studies on no-tension and no-compression solids have been found, the variational principle constructing the cornerstone of elasticity is not studied thoroughly. The paper presents two concise variational formulations, a principle of minimum potential energy and a principle of minimum complementary energy, which are available both for no-tension and no-compression solids. Linearization of the conic yield surfaces leads to a series of linear complementary constitutive equations that are embedded into the proposed variational framework. Differing from other variational formulations, an approximate total solution rather than the Newton iteration is achieved in finite element analysis. It makes the algorithm stable. The applications include a no-tension panel benchmark test, two masonry structures and a wrinkled membrane. Compared with our previous study on bi-modulus materials, the newly developed variational formulation is capable of capturing the evolution of wrinkles in membranes, and can be used for the analysis and design of wrinkle-free structures.

How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational framework

Three regularization concepts are assessed regarding their variational structure and interference with the predicted physics of (quasi-)brittle damage: the fracture energy concept, viscous regularization and micromorphic regularization. They are first introduced in a unified variational framework, depicting how they distinctively evolve from incremental energy minimization. The analysis of a certain time interval of a one-dimensional example is used to show how viscous and micromorphic regularization retains well-posedness within the softening regime. By way of contrast, the fracture energy concept is characterized by ill-posedness—as known from previous non-variational analyses. Numerical examples finally demonstrate the limitations and capabilities of each concept. The ill-posed local fracture energy concept leads by its design to a spatially constant fracture energy—in line with Griffith’s theory. The viscous regularization, in turn, yields a well-posed problem but artificial viscosity can add a bias to unloading and fracture thickness. Furthermore, and even more important, a viscous regularization does not predict a spatially constant fracture energy due to locally heterogeneous loading rates. The well-posed micromorphic regularization is in line with the underlying physics and does not show this undesired dependency. However, it requires the largest numerical efforts, since it is based on a coupled two-field formulation.

VITAS: A multi-purpose simulation code for the solution of neutron transport problems based on variational nodal methods

Existing neutronic whole-core simulation codes are problem-oriented. With the advancement of computer technology and the development of complicated micro-nuclear reactors, there is a pressing need to create solution methods with broad applicability and the ability to accommodate more complex, high-fidelity, unstructured geometries. This necessity motivates the development of a neutron transport code that can adapt to different types of nuclear reactors. This work describes the code design and theoretical model of VITAS, a multi-purpose simulation code for handling steady-state and time-dependent neutron transport problems. This code incorporates algorithms and models inside a unified variational nodal framework based on the second-order neutron transport equation. The most recent version of VITAS employs the isotropic scattering approximation and permits arbitrary spatial and angular refinement. VITAS is compatible with Cartesian geometry, hexagonal geometry, triangular geometry, and Cartesian geometry with finite element subdivisions in the radial direction (Cartesian-FE). The three-dimensional modeling can be performed by axially extruding the two-dimensional mesh. Users can choose the mesh type and expansion order on-demand for optimal computational efficiency.In this paper, VITAS is validated using benchmark problems, i.e., the TAKEDA3 benchmark and the transient LMW benchmark with Cartesian homogenized geometry, the C5G7 and C5G7-TD exercises with Cartesian-FE geometry, the TAKEDA4 benchmark and the Buckner and Stewart benchmark with hexagonal-z homogenized geometry, and the Dodds benchmark with cylindrical-z geometry described by triangular-z unstructured meshes. The space-angle truncation is extensively examined to demonstrate the asymptotic convergence behavior with p- and h-refinement. The results indicate that VITAS has the potential to become a versatile toolset for solving a variety of nuclear reactor problems.

A Unified Pansharpening Method With Structure Tensor Driven Spatial Consistency and Deep Plug-and-Play Priors

Pansharpening is to generate a high resolution multispectral (HRMS) image by preserving the spectral information from a low resolution multispectral (LRMS) image and the spatial content from a panchromatic (PAN) image. This article proposes a unified pansharpening method with structure tensor driven spatial consistency and deep plug and play priors. First, the spectral fidelity constraint between HRMS and LRMS is imposed for preserving spectral information. Second, the structure tensor is applied to characterize the spatial geometric information of HRMS and PAN images, thus the structure tensor driven spatial consistency prior between HRMS and PAN is particularly exploited for preserving spatial content. Moreover, by generalizing the convolution neural network (CNN) fusion method into a unified variational framework, a novel CNN-based deep plug and play prior between the HRMS and CNN-based fused MS images is also proposed to generate more image characteristics for further preserving spectral information and spatial content. Besides, the proposed model is solved by the alternating direction method of multipliers (ADMM) algorithm. Finally, extensive experiments on both reduced and full resolution by comparing with various representative approaches exhibit the excellent performance of the proposed method.

Bridging the gap between local and nonlocal numerical methods—A unified variational framework for non-ordinary state-based peridynamics

The paper aims to develop a unified variational framework to bridge the gap between the non-ordinary state-based peridynamics (NOSB-PD) and the classical continuum mechanics (CCM). First, a new force state vector is proposed by introducing the first Piola–Kirchhoff stress. This new force state vector enables the stress divergence of each material point to be expressed by averaging all the force state vectors in its support domain. The new force state vector also ensures the mathematical consistency between the strong form of PD and CCM when the horizon of a material point approaches to zero. Second, the displacement and traction boundaries in CCM are transformed into the non-local fictious boundary layers in PD, and a non-local Gauss’s formulation is presented by transforming the displacement and traction boundaries in CCM into the non-local fictious boundary layers in PD, and this formulation unifies the variational framework and boundary conditions of PD and CCM. Third, a fully implicit algorithm is developed to obtain the general nonlinear problems such as fracture and large deformation of solid materials. Further, a penalty method is employed to eliminate the zero-energy mode oscillation inherently observed in NOSB-PD, and the penalty force and penalty stiffness matrix are derived for the proposed implicit algorithm and numerical implementation. Numerical results demonstrate that the proposed method is accurate and can well capture the fracture and large deformation of solid materials. Results also indicate that the method can effectively prevent the zero-mode oscillations inherently observed in the original NOSB-PD, and thus ensures the computational stability.

A two species micro–macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach

Wormlike micelles are self-assemblies of polymer chains that can break and recombine reversibly. In this paper, we derive a thermodynamically consistent two-species micro–macro model of wormlike micellar solutions by employing an energetic variational approach. The model incorporates a breakage and combination process of polymer chains into the classical micro–macro dumbbell model of polymeric fluids in a unified variational framework. We also study different maximum entropy closure approximations to the micro-macro model by “variation-then-closure” and “closure-then-variation” approaches. By imposing a proper dissipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”, preserves the thermodynamical structure of both mechanical and chemical parts of the original system. Several numerical examples show that the closure model can capture the key rheological features of wormlike micellar solutions in shear flows.

Numerical homogenization beyond scale separation

Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

Modeling of Covid‐19 trade measures on essential products: a multiproduct, multicountry spatial price equilibrium framework

In this paper, we develop a unified variational inequality framework in the context of spatial price network equilibrium problems that handles multiple products with multiple demand and supply markets in multiple countries as well as multiple transportation routes. The model incorporates a plethora of distinct trade measures, which is particularly important in the pandemic, as PPEs and other essential products are in high demand, but short in supply globally. In the model, product flows as well as prices at the supply markets and the demand markets in different countries are variables that allows us to seamlessly introduce various trade measures, including tariffs, quotas, as well as price floors and ceilings. Qualitative properties are analyzed. Numerical examples are provided to illustrate the impacts of the trade measures on equilibrium product path and link flows, and on prices, and demand and supply quantities. Given the relevance of the trade measures in the world today and discussions concerning the impacts, the framework constructed in this paper is especially timely.

Variational generalization of the Green–Naghdi and Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates

This paper derives an averaged Lagrangian functional for dynamic coupling between rigid-body motion and its interior shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations for the interior fluid sloshing with 3-D rotation vector and translations, and also the equations of motion for the linear momentum and angular momentum of the rigid-body containing shallow water, are derived from the averaged Lagrangian functional, which describes a columnar motion, by using Hamilton’s principle and the Euler–Poincaré variational framework. The generalized Green–Naghdi equations have a form of potential vorticity (PV) conservation, which can be obtained from the particle-relabelling symmetry, and is a combination of the PV derived by Miles and Salmon (1985) and the PV derived by Dellar and Salmon (2005) for geophysical fluid dynamics problems, where the rotation vector varies spatially. By applying the assumption of zero-potential-vorticity flow to the averaged Lagrangian functional, a new set of Boussinesq-like evolution equations are derived, which are a generalization of the Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates. Moreover, the new variational principles are appended to Luke’s variational principle to present a unified variational framework for the hydrodynamic problem of interactions between gravity-driven potential-flow water waves and a freely floating rigid-body, dynamically coupled to its interior weakly dispersive nonlinear shallow-water sloshing in three dimensions.

Joint Analysis and Weighted Synthesis Sparsity Priors for Simultaneous Denoising and Destriping Optical Remote Sensing Images

Stripe and random noise are two different degradation phenomena that commonly coexist in optical remote sensing images, and they are often modeled as inverse problems. In model-based inverse problems, analysis and synthesis sparse representations (SSRs) are used as regularization terms to obtain approximate solutions due to their respective merits, i.e., the nonzero coefficients in SSR are usually used to describe an image, while the indexes of zeros in analysis sparse representation (ASR) are used to characterize the stripe. Inspired by these merits, we propose a unified variational framework, called a joint analysis and weighted synthesis (JAWS) sparsity model, to simultaneously separate the clean image and the stripe from a single optical remote sensing image. To solve the JAWS sparsity model efficiently, an alternating minimization optimization strategy is first employed to separate it into two subproblems that are used for different tasks. One called as weighted SSR (WSSR) is the main for optical remote sensing image denoising, which can be effectively solved by employing the weighted singular value thresholding operator, while the other called as ASR is the main approach for optical remote sensing image destriping, which is optimized by adopting the split Bregman iteration. By minimizing the two subproblems alternatively, the proposed JAWS sparsity model is efficiently solved. Finally, both quantitative and qualitative results of experiments on synthetic and real-world optical remote sensing images validate that the proposed approach is effective and even better than the state of the arts.

A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.

Tariffs and quotas in world trade: A unified variational inequality framework

In this paper, we develop a unified variational inequality framework in the context of spatial price network equilibrium problems that captures world trade policies in the form of tariff rate quotas, which are two-tiered tariffs, and that have been applied in practice to numerous commodities. The spatial price network equilibrium model allows for multiple supply markets and multiple demand markets in different countries as well as multiple transportation routes. The model is qualitatively analyzed and also related to models with trade policies such as ad valorem tariffs and strict quotas. A case study on the dairy industry focusing on the United States and France is presented to illustrate the modeling and computational framework with impacts of stricter quotas, higher over quota tariffs, as well as additional transportation routes assessed. The numerical examples reveal that although domestic producers can gain under tariff rate quotas, consumers may experience higher prices. Given the relevance of tariffs to global trade in the world today and discussions concerning the impacts, the framework constructed in this paper is especially timely.

Whiteness Constraints in a Unified Variational Framework for Image Restoration

We propose a robust variational model for the restoration of images corrupted by blur and the general class of additive white noises. The key idea behind our proposal relies on a novel hard constra...

Whiteness Constraints in a Unified Variational Framework for Image Restoration

We propose a robust variational model for the restoration of images corrupted by blur and the general class of additive white noises. The key idea behind our proposal relies on a novel hard constraint imposed on the residual of the restoration, namely we characterize a residual whiteness set to which the restored image must belong. As the feasible set is unbounded, solution existence results for the proposed variational model are given. Moreover, based on theoretical derivations as well as on Monte Carlo simulations, we provide well-founded guidelines for setting the whiteness constraint limits. The solution of the non-trivial optimization problem, due to the non-smooth non-convex proposed model, is efficiently obtained by an alternating directions method of multipliers, which in particular reduces the solution to a sequence of convex optimization subproblems. Numerical results show the potentiality of the proposed model for restoring blurred images corrupted by several kinds of additive white noises.

Cluster Sparsity Field: An Internal Hyperspectral Imagery Prior for Reconstruction

Hyperspectral images (HSIs) have significant advantages over more traditional image types for a variety of computer vision applications dues to the extra information available. The practical reality of capturing and transmitting HSIs however, means that they often exhibit large amounts of noise, or are undersampled to reduce the data volume. Methods for combating such image corruption are thus critical to many HSIs applications. Here we devise a novel cluster sparsity field (CSF) based HSI reconstruction framework which explicitly models both the intrinsic correlation between measurements within the spectrum for a particular pixel, and the similarity between pixels due to the spatial structure of the HSI. These two priors have been shown to be effective previously, but have been always considered separately. By dividing pixels of the HSI into a group of spatial clusters on the basis of spectrum characteristics, we define CSF, a Markov random field based prior. In CSF, a structured sparsity potential models the correlation between measurements within each spectrum, and a graph structure potential models the similarity between pixels in each spatial cluster. Then, we integrate the CSF prior learning and image reconstruction into a unified variational framework for optimization, which makes the CSF prior image-specific, and robust to noise. It also results in more accurate image reconstruction compared with existing HSI reconstruction methods, thus combating the effects of noise corruption or undersampling. Extensive experiments on HSI denoising and HSI compressive sensing demonstrate the effectiveness of the proposed method.

A unified variational eigen-erosion framework for interacting brittle fractures and compaction bands in fluid-infiltrating porous media

The onset and propagation of the cracks and compaction bands, and the interactions between them in the host matrix, are important for numerous engineering applications, such as hydraulic fracture and CO2 storage. While crack may become flow conduit that leads to leakage, formation of compaction band often accompanies significant porosity reduction that prevents fluid to flow through. The objective of this paper is to present a new unified framework that predicts both the onset, propagation and interactions among cracks and compaction bands via an eigen-deformation approach. By extending the generalized Griffith’s theory, we formulate a unified nonlocal scheme that is capable to predict the fluid-driven fracture and compression-driven anti-crack growth while incorporating the cubic law to replicate the induced anisotropic permeability changes triggered by crack and anti-crack growth. A set of numerical experiments are used to demonstrate the robustness and efficiency of the proposed model.