Variational Formulation Research Articles

Directional flow in perivascular networks: mixed finite elements for reduced-dimensional models on graphs

AbstractFlow of cerebrospinal fluid through perivascular pathways in and around the brain may play a crucial role in brain metabolite clearance. While the driving forces of such flows remain enigmatic, experiments have shown that pulsatility is central. In this work, we present a novel network model for simulating pulsatile fluid flow in perivascular networks, taking the form of a system of Stokes–Brinkman equations posed over a perivascular graph. We apply this model to study physiological questions concerning the mechanisms governing perivascular fluid flow in branching vascular networks. Notably, our findings reveal that even long wavelength arterial pulsations can induce directional flow in asymmetric, branching perivascular networks. In addition, we establish fundamental mathematical and numerical properties of these Stokes–Brinkman network models, with particular attention to increasing graph order and complexity. By introducing weighted norms, we show the well-posedness and stability of primal and dual variational formulations of these equations, and that of mixed finite element discretizations.

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

This paper is concerned with the numerical solution of the flow problem in a fractured porous medium where the fracture is treated as a lower dimensional object embedded in the rock matrix. We consider a space-time mixed variational formulation of such a reduced fracture model with mixed finite element approximations in space and discontinuous Galerkin discretization in time. Different spatial and temporal grids are used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. Analysis of the numerical scheme, including well-posedness of the discrete problem, stability and a priori error estimates, is presented. Using substructuring techniques, the coupled subdomain and fracture system is reduced to a space-time interface problem which is solved iteratively by GMRES. Each GMRES iteration involves solution of time-dependent problems in the subdomains using the method of lines with local spatial and temporal discretizations. The convergence of GMRES is proved by using the field-of-values analysis and the properties of the discrete space-time interface operator. Numerical experiments are carried out to illustrate the performance of the proposed iterative algorithm and the accuracy of the numerical solution.

A study of sign-changing Poisson-type equations in two configurations

Abstract. In this paper, We are interested in specific non-coercive issues concerning electromagnetic wave propagation in the presence of metals or particular metamaterials. We focus on some non-coercive problems which cannot be studied by a classical Lax-Milgram approach. We consider the sign-changing Poisson-type equations with the homogeneous Dirichlet boundary conditions in the circular configuration and in the three square configuration. We utilize a method called T-isomorphism, which allows the conversion of non-coercive problems into coercive ones, to examine specific Poisson-type equations with sign-changing parameters. We first use the classical method to transform the equations into the corresponding variational formulations, and then apply the T-isomorphism method to show the well-posedness of these variational formulations with some parameters. By applying appropriate isomorphisms, we can conclude the well-posedness of the related variational formulations with certain parameters, which constitute the main findings of this paper. After giving appropriate isomorphisms, we can deduce the well-posedness of the corresponding variational formulations with some parameters, which are our main results in this paper.

Static and dynamic aspects of the principle of virtual work

AbstractThe paper presents a detailed analysis of the principle of virtual work in its statics and dynamics aspects. Special attention is paid to not exact formulation and interpretation of mechanical system motion equations based upon this principle, what is presented quite often in text books. The general form of the principle of virtual work and variational formulations of the problems in statics are presented and analyzed. Additionally, the dynamic implications of the principle of virtual work and dynamics problem formulation of a system subjected to scleronomic constraints are detailed. The basis for the formulation is the dynamic implication of the principle of virtual work that specifies the relation between the constraint reactions and the body accelerations. This relation takes the form of the variation inequality. As the result, the Gauss principle which determines the body acceleration vector is presented. Furthermore, a description of impact formulation, based upon the Carnot model, caused by the constraints is provided. The theoretical development is illustrated with examples of applications of the principle of virtual work.

Using k-means to sort spectra: Electronic order mapping from scanning tunneling spectroscopy measurements

Hyperspectral imaging techniques have a unique ability to probe the inhomogeneity of material properties whether driven by compositional variation or other forms of phase segregation. In the doped cuprates, iridates, and related materials, scanning tunneling microscopy/spectroscopy (STM/STS) measurements have found the emergence of pseudogap “puddles” from the macroscopically Mott insulating phase with increased doping. However, categorizing this hyperspectral data by electronic order is not trivial and has often been done with ad hoc methods. In this paper, we demonstrate the utility of k-means, a simple and easy-to-use unsupervised clustering method, as a tool for classifying heterogeneous scanning tunneling spectroscopy data by electronic order for Rh-doped Sr2IrO4, a cuprate-like material. Applied to STM data acquired within the Mott phase, k-means was able to identify areas of Mott order and of pseudogap order. The unsupervised nature of k-means limits avenues for bias and provides clustered spectral shapes without a priori knowledge of the physics. Additionally, we demonstrate the use of k-means as a preprocessing tool to constrain phenomenological function fitting. Clustering the data allows us to reduce the fitting parameter space, limiting over-fitting. We suggest k-means as a fast, simple model for processing hyperspectral data on materials of mixed electronic order.

Denoising sphere‐valued data by relaxed total variation regularization

AbstractCircle‐ and sphere‐valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore ‐sphere‐valued signals exploiting the classical anisotropic total variation on the surrounding ‐dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non‐convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0‐sphere‐valued) signals, the relaxation is provable tight, that is, the relaxed solution can be used to construct a solution of the original non‐convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)‐valued signals.

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Both Effective Elastic Properties and Voids Interaction Energy Estimations within the Context of 3D Nano-Porous Materials

This study aims to estimate elastic properties as well as voids interaction energy within the context of 3D nano-porous materials exhibiting spherical or cylindrical voids, with or without the effect of surface energy. For this purpose, numerical homogenization is applied, based on the finite element method (FEM) in combination with the level-set technique. Surface energy is approximated by explicitly taking into account its contribution to overall stiffness in the variational formulation. Several cases are then investigated to appreciate the interaction energy between multiple 3D nanovoids on the actual behavior of nanomaterials. As expected, and like surface energy, the results show that interaction energy keeps affecting the effective behavior by increasing the size of RVE as well as the number of voids inside. This influence is more marked the more cylindrical the voids and the more closely they are adjacent to each other.

Constraints on Female Labor Force Participation in Qatar: Understanding the Effects of Marriage, Family, and Traditional Values

Abstract This paper examines constraints to female labor force participation in Qatar, a country where state revenue from natural resource rents means any citizen can secure a public sector job should they seek employment. Using data from an original survey and focus groups, we find that although Qataris are generally in support of female labor force participation – both in principle and for their own relatives – concerns remain about the impact of women working for marriage, family, and traditional values. These concerns may be especially salient for younger Qatari men – a constituency particularly impacted by erosion of traditional gender norms at a time when they are establishing their own households. We also find that Qataris with a larger percentage of male children tend to be less supportive of women working outside of the home, suggesting forms of family-level variation in investment in patriarchal values and child rearing practices.

Conservation laws of the one-dimensional relativistic magnetogasdynamics equations

Abstract The present paper offers a comprehensive analysis of the one-dimensional relativistic magnetogasdynamics equations in Lagrangian coordinates. These equations, describing the behavior of a relativistic magnetized gas, are reformulated in variational form to facilitate further analysis. A complete group analysis of the resulting Euler-Lagrange equation is performed, allowing us to systematically identify the symmetries inherent in the system. Using Noether’s theorem, we derive conservation laws in Lagrangian coordinates, offering critical insights into the invariants and conserved quantities of the system. This work expands the theoretical understanding of relativistic magnetogasdynamics.

Geophysical effects caused by the bolide fall on September 02, 2023 (Turkey)

We present the results of instrumental observations of acoustic oscillations, geomagnetic variations and variations of the atmospheric electric field during the fall and explosive destruction of the bolide in the southeast of Turkey on September 02, 2023. It is shown that the destruction of the bolide under the action of aerodynamic forces, which occurred in three stages, was accompanied by an acoustic signal of a characteristic shape and manifested in variations of the magnetic and electric fields in the near-surface layer atmosphere. The total energy of the event, estimated by the acoustic effect, was ~ 9×1012 J, which corresponds to about 2.15 kt in TNT equivalent. The maximum amplitude of geomagnetic variations caused by the explosion of the bolide ranged from 0.2 to 2.1 nT depending on the distance. At the same time, the amplitude of variations of the vertical component of the atmospheric electric field at the Mikhnevo observatory (distance ~1900 km) was ~70 V/m. The ionospheric effect of the event under consideration is demonstrated in the form of variations of the critical frequency f0F2 obtained as a result of processing ionograms of height-frequency sounding of the ionosphere at the Rome station.

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

ABSTRACTIn this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

First Principles Variational Formulation and Analytical Solutions for Third Order Shear Deformable Beams with Simple Supports using Fourier Series Method

The Fourier series method for solving bending problems of third-order shear deformable beams (TSBD) is presented in this paper. The theory accounts for transverse shear deformation and is suitable for moderately thick beams. Transverse shear stress-free conditions are valid at the beam surfaces for TSDB. The field equations are two coupled ordinary differential equations in terms of two unknown displacement functions – transverse deflection w(x) and warping function For simply supported ends considered, the loading and unknown functions w(x) and are represented in Fourier series that satisfies the boundary conditions. The problem is reduced to a system of algebraic equations in terms of the Fourier coefficients wn and which is solved to obtain wn and Axial and transverse displacements fields, axial bending stress and transverse shear stress are then determined for the cases of point load at midspan, uniformly and linearly distributed loads over the span. The results are identical with previous results obtained by other scholars who used Ritz variational methods and other mathematical tools. For moderately thick beams under uniform load with l/h = 4, the results obtained for is 0.516% greater than the exact result by Timoshenko and Goodier while for thick beams under uniform load with l/h = 2, the result for is 1.96% greater than the exact result by Timoshenko and Goodier. Similar acceptable variation was obtained for for beams under linearly distributed load with the variations being less than 2% for l/h = 2. However, unacceptable variations of 68.37% were found for in thick beams under point load, for l/h = 2.0. The variation for however reduced to 12.513% for moderately thick beams under point load for l/h =4.

On the bending, buckling and free vibration analysis of bio-inspired helicoidal laminated composite shear and normal deformable beams

The mechanical behaviours of bio-inspired helicoidal symmetric laminated composite (BIHLC) beams are investigated via the Ritz method. By exploiting the variational formulation, equations of motion along with element stiffness, geometrical stiffness, and mass matrices are derived. The study conducts a thorough examination, covering bending, buckling stability, and free vibration analyses of BIHLC beams with various lamination schemes. The developed model is verified against existing literature on conventional composite laminated and BIHLC beams. The study also examines the mechanical response of BIHLCs, considering boundary conditions, lamination schemes, orthotropy ratios, and aspect ratios. Notably, deflections, critical buckling loads, and fundamental frequencies demonstrate variations dependent on the specific lamination scheme, boundary condition, and aspect ratio. Novel findings, presented for the first time, offer valuable insights for future studies in this area.

Variational Analysis of a Dynamic Contact Problem with Wear and Damage Involving Viscoelastic Materials with Long-Term Memory

We investigate a mathematical problem involving dynamic interaction between a viscoelastic body with long-term memory loss and an obstruction. The contact is frictional and bilateral, with a moving rigid base, resulting in wear of the contacting surface. The problem is expressed as a coupled system, with a hyperbolic quasi-variational inequality for displacement and a parabolic variational inequality for damage. We define a variational formulation for the model and demonstrate the existence of a single weak solution to the problem. The material behaviour is explained using a viscoelastic constitutive law that includes long-term memory and damage. Elastic deformations induce material deterioration, which is depicted by a parabolic inclusion. The proof is based on classical existence.

A Comparative Study for Evaluating the Groundwater Inflow and Drainage Effect of Jinzhai Pumped Storage Power Station, China

Various hydrogeological problems like groundwater inflow, water table drawdown, and water pressure redistribution may be encountered in the construction of hydraulic projects. How to accurately predict the occurrence of groundwater inflow and assess the drainage effect during construction are still challenging problems for engineering designers. Taking the Jinzhai pumped storage power station (JPSPS) of China as an example, this paper aims to use different methods to calculate the water inflow rates of an underground powerhouse and evaluate the drainage effect caused by tunnel inflow during construction. The methods consist of the analytical formulas, the site groundwater rating (SGR) method, and the Signorini type variational inequality formulation. The results show that the analytical methods considering stable water table may overestimate the water inflow rates of caverns in drained conditions, whereas the SGR method with available hydro-geological parameters obtains a qualitative hazard assessment in the preliminary phase. The numerical solutions provide more precise and reliable values of groundwater inflow considering complex geological structures and seepage control measures. Moreover, the drainage effects, including a seepage-free surface, pore water pressure redistribution, and hydraulic gradient, have been accurately evaluated using various numerical synthetic cases. Specifically, the faults intersecting on underground caverns and drainage structures significantly change the groundwater flow regime around caverns. This comparative study can not only exactly identify the capabilities of the methods for cavern inflow in drained conditions, but also can comprehensively evaluate the drainage effect during cavern construction.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Model Order Reduction of an Ultraweak and Optimally Stable Variational Formulation for Parametrized Reactive Transport Problems

Model Order Reduction of an Ultraweak and Optimally Stable Variational Formulation for Parametrized Reactive Transport Problems

Viromes vs. mixed community metagenomes: choice of method dictates interpretation of viral community ecology

BackgroundViruses, the majority of which are uncultivated, are among the most abundant biological entities on Earth. From altering microbial physiology to driving community dynamics, viruses are fundamental members of microbiomes. While the number of studies leveraging viral metagenomics (viromics) for studying uncultivated viruses is growing, standards for viromics research are lacking. Viromics can utilize computational discovery of viruses from total metagenomes of all community members (hereafter metagenomes) or use physical separation of virus-specific fractions (hereafter viromes). However, differences in the recovery and interpretation of viruses from metagenomes and viromes obtained from the same samples remain understudied.ResultsHere, we compare viral communities from paired viromes and metagenomes obtained from 60 diverse samples across human gut, soil, freshwater, and marine ecosystems. Overall, viral communities obtained from viromes had greater species richness and total viral genome abundances than those obtained from metagenomes, although there were some exceptions. Despite this, metagenomes still contained many viral genomes not detected in viromes. We also found notable differences in the predicted lytic state of viruses detected in viromes vs metagenomes at the time of sequencing. Other forms of variation observed include genome presence/absence, genome quality, and encoded protein content between viromes and metagenomes, but the magnitude of these differences varied by environment.ConclusionsOverall, our results show that the choice of method can lead to differing interpretations of viral community ecology. We suggest that the choice of whether to target a metagenome or virome to study viral communities should be dependent on the environmental context and ecological questions being asked. However, our overall recommendation to researchers investigating viral ecology and evolution is to pair both approaches to maximize their respective benefits.ADm-mTTAXM3aVZwCyoroCBVideo

A cluster-based incremental potential approach for reduced order homogenization of bones.

We develop a cluster-based model order reduction (called C-pRBMOR) approach for efficient homogenization of bones, compatible with a large variety of generalized standard material (GSM) models. To this end, the pRBMOR approach based on a mixed incremental potential formulation is extended to a clustered version for a significantly improved computational efficiency. The microscopic modeling of bones falls into a mixed incremental class of the GSM framework, originating from two potentials. An offline phase of the C-pRBMOR approach includes both a clustering analysis spatially decomposing the micro-domain within an RVE and a space-time decomposition of the microscopic plastic strain fields. A comparative study on two different clustering approaches and two algorithms for mode identification is additionally conducted. For an online analysis, a cluster-enhanced version of evolution equations for the reduced variables is derived from an effective incremental variational formulation, rendering a very small set of nonlinear equations to be numerically solved. Several numerical examples show the effectiveness of the C-pRBMOR approach. A striking acceleration rate beyond 104 against conventional FE computations and that beyond 103 against the original pRBMOR approach are observed.