Multiplier Field Research Articles

Casimir energy with perfect electromagnetic boundary conditions and duality: A field-theoretic approach

Using functional integral methods, we study the Casimir effect for the case of two infinite parallel plates in the QED vacuum, with (different) perfect electromagnetic boundary conditions applied to both plates. To enforce these boundary conditions, we add two Lagrange multiplier fields to the action. We subsequently recover the known Casimir energy in two ways: once directly from the path integral and once as the vacuum expectation value of the 00 component of the energy-momentum tensor. Comparing both methods, we show that the energy-momentum tensor must be modified and that it picks up boundary contributions as a consequence. We also discuss electromagnetic duality invariance of the theory and its interplay with the boundaries by generalizing the Deser-Teitelboim implementation of the duality transformation. Published by the American Physical Society 2024

Propagation of gravitational waves in a dynamical wormhole background for two-scalar Einstein–Gauss–Bonnet theory

In this work, we propose a model of Einstein–Gauss–Bonnet gravity coupled with two scalar fields. The two scalar fields are considered to be “frozen” or they become non-dynamical by employing appropriate constraints in terms of Lagrange multiplier fields. We show that, even in the case that the arbitrary spherically symmetric spacetime is dynamical, we can construct a model where the wormhole spacetime is a stable solution in this framework. We especially concentrate on the model reproducing the dynamical wormhole, where the wormhole appears in a finite-time interval. We investigate the propagation of the gravitational wave in the wormhole spacetime background and we show that the propagation speed is different from that of light in general, and there is a difference in the speeds between the incoming propagating wave and the outgoing propagating gravitational wave. We need to stress that the wormhole model used for exploring gravitational wave propagation is not obtained by solving any equations but is essentially constructed by hand.

Wormholes inside stars and black holes

We construct models of two exotic objects: (i) a wormhole whose throat is hidden by a stellar object like a neutron star; and (ii) a wormhole inside a black hole. We work within Einstein’s gravity coupled to two scalar fields with a specific choice of the scalar field Lagrangian. In general, the model contains ghosts, but they are eliminated using the constraints given by the Lagrange multiplier fields. The constraints are a generalization of the mimetic constraint, where nondynamical dark matter effectively appears. As a result, in our model, instead of the nondynamical dark matter, nondynamical exotic matter like a phantom effectively arises. For the mixed wormhole-plus-star system, we find the corresponding mass-radius relation and show that it is possible to get characteristics comparable to those of ordinary neutron stars. For the wormhole inside the black hole, we find an extremal limit where the radius of the throat coincides with the radius of the event horizon and demonstrate that the Hawking temperature vanishes in this limit. Published by the American Physical Society 2024

Field redefinition invariant Lagrange multiplier formalism with gauge symmetries

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant.

Supercomputers against strong coupling in gravity with curvature and torsion

Many theories of gravity are spoiled by strongly coupled modes: the high computational cost of Hamiltonian analysis can obstruct the identification of these modes. A computer algebra implementation of the Hamiltonian constraint algorithm for curvature and torsion theories is presented. These non-Riemannian or Poincaré gauge theories suffer notoriously from strong coupling. The implementation forms a package (the ‘Hamiltonian Gauge Gravity Surveyor’ – HiGGS) for the xAct tensor manipulation suite in Mathematica. Poisson brackets can be evaluated in parallel, meaning that Hamiltonian analysis can be done on silicon, and at scale. Accordingly HiGGS is designed to survey the whole Lagrangian space with high-performance computing resources (clusters and supercomputers). To demonstrate this, the space of ‘outlawed’ Poincaré gauge theories is surveyed, in which a massive parity-even/odd vector or parity-odd tensor torsion particle accompanies the usual graviton. The survey spans possible configurations of teleparallel-style multiplier fields which might be used to kill-off the strongly coupled modes, with the results to be analysed in subsequent work. All brackets between the known primary and secondary constraints of all theories are made available for future study. Demonstrations are also given for using HiGGS – on a desktop computer – to run the Dirac–Bergmann algorithm on specific theories, such as Einstein–Cartan theory and its minimal extensions.

Ultrasonic characterization of distributed cracking damage in concrete using multiply scattered wave fields

This paper reports work to detect, visualize, and characterize alkali–silica reactivity cracking damage in concrete using multiply scattered ultrasonic wave fields. Numerical simulations and laboratory-scale experiments are performed to understand ultrasonic surface wave scattering caused by distributed cracks in concrete, distinguishing that caused by the internal aggregate network. The simulations and experimental results reveal that incident ultrasonic surface waves undergo complicated multiple scattering set up by distributed cracks. To extract the crack-induced multiply scattered ultrasonic wave fields, a frequency-wavenumber (f-k) domain signal filtering approach is proposed. The feasibility of the signal analysis approach is then established by a series of experiments on large steel reinforced concrete blocks with known and controlled cracking damage. The experimental results demonstrate that distributed cracks can be detected and visualized using the proposed ultrasonic signal analysis approach, even at very early stages of the damage process. In addition, we demonstrate that the progress of damage over time can be monitored, where the cracking damage extent is closely related to the extracted multiply scattered wave field energy.

An interface-enriched generalized finite element formulation for locking-free coupling of non-conforming discretizations and contact

We propose an enriched finite element formulation to address the computational modeling of contact problems and the coupling of non-conforming discretizations in the small deformation setting. The displacement field is augmented by enriched terms that are associated with generalized degrees of freedom collocated along non-conforming interfaces or contact surfaces. The enrichment strategy effectively produces an enriched node-to-node discretization that can be used with any constraint enforcement criterion; this is demonstrated with both multi-point constraints and Lagrange multipliers, the latter in a generalized Newton implementation where both primal and Lagrange multiplier fields are updated simultaneously. We show that the node-to-node enrichment ensures continuity of the displacement field—without locking—in mesh coupling problems, and that tractions are transferred accurately at contact interfaces without the need for stabilization. We also show the formulation is stable with respect to the condition number of the stiffness matrix by using a simple Jacobi-like diagonal preconditioner.

Black holes with Lagrange multiplier and potentialin mimetic-like gravitational theory: multi-horizon black holes

In this paper, we consider the mimetic-like field equations coupled with the Lagrange multiplier and the potential to derive non-trivial spherically symmetric black hole (BH) solutions. We divided this study into three cases: in the first one, we choose the Lagrange multiplier and the potential to vanish and derive a BH solution that coincides with the BH of the Einstein general relativity despite the non-vanishing value of the mimetic-like scalar field. The first case is consistent with the previous studies in the literature where the mimetic theory coincides with GR [1]. In the second case, we derive a solution with a constant value of the potential and a dynamical value of the Lagrange multiplier. This solution has no horizon, and therefore, the obtained space-time does not correspond to the BH. In this solution, there appears a region of the Euclidian signature where the signature of the diagonal components of the metric is (+,+,+,+) or the region with two times where the signature is (+,+,-,-). Finally, we derive a BH solution with non-vanishing values of the Lagrange multiplier, potential, and mimetic-like scalar field. This BH shows a soft singularity compared with the Einstein BH solution. The relevant physics of the third case is discussed by showing their behavior of the metric potential at infinity, calculating their energy conditions, and studying their thermodynamical quantities. We give a brief discussion on how our third case can generate a BH with three horizons as in the de Sitter-Reissner-Nordström black hole space-time, where the largest horizon is the cosmological one and two correspond to the outer and inner horizons of the BH. Even in the third case, the region of the Euclidian signature or the region with two times appears. We give a condition that such unphysical region(s) is hidden inside the black hole horizon and the existence of the region(s) becomes less unphysical. We also study the thermodynamics of the multi-horizon BH and consider the extremal case, where the radii of two horizons coincide with each other. We observe that the Hawking temperature and the heat capacity vanish in the extremal limit. Finally, we would like to stress the fact that in spite that the field equations we use have no cosmological constant, our BH solutions of the second and third case behave asymptotically as AdS/dS.

Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

The present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a L_2-optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

Thermal gauge theories with Lagrange multiplier fields

We study the Yang–Mills theory and quantum gravity at finite temperature, in the presence of Lagrange multiplier fields. These restrict the path integrals to field configurations, which obey the classical equations of motion. This has the effect of doubling the usual one-loop thermal contributions and of suppressing all radiative corrections at higher-loop order. Such theories are renormalizable at all temperatures. Some consequences of this result in quantum gravity are briefly examined.

Restricting loop expansions in gauge theories coupled to matter

Quantizing any model in which a Lagrange multiplier (LM) field is used to restrict field configurations to those that satisfy the classical equations of motion, leads to at most one-loop radiative corrections. This approach can be used with both the Yang–Mills (YM) and Einstein–Hilbert (EH) action; the resulting theory is both renormalizable and unitary, has a positive energy spectrum and has no negative norm states contributing to physical processes. Although this approach cannot be consistently used with scalar fields alone, scalar fields can be coupled to gauge fields so that loop effects in the gauge sector are restricted to one-loop order in a way that satisfies the usual criterion for a consistent quantum field theory. The tree-level diagrams are those of the classical theory in which the metric couples to the energy–momentum tensor.

Assessing the dynamic tourism inter-industry linkages and economic structural changes in Cambodia\u2019s economy

This study employs the SAM-based model combining with multiplier product matrix and field of influence approaches. Under three input–output transaction table matrices of the years 2005, 2010, and 2015, these approaches assess the dynamic tourism inter-industry linkages and structural economic changes in Cambodia. We find that the overall inter-industry connection is relatively low. The textile, other manufacturing, and transportation and communication are key sectors. They have the largest coefficient field of influence of changes in the economic system. Tourism has shifted to be a key sector in 2010 and 2015. However, its backward and forward linkages are still small. It is a relatively promising sector generating a large coefficient field of influence of changes, showing less strength of overall connection with other industries. This study may suggest that there would be a need for promoting, encouraging, and investing in key economic sectors. Policy intervention should focus on developing domestic tourism linkages and strengthening inter-industry ties to diversity tourism benefits the local economy.

A non-incremental numerical method for dynamic elastoplastic problems by the symplectic Brezis–Ekeland–Nayroles principle

The goal of the present work is the demonstration of the ability of the Symplectic Brezis–Ekeland–Nayroles (SBEN) principle for the numerical simulation of dynamic elastoplastic problems in small strains. This non-incremental approach is an alternative method to conventional step-by-step techniques, as it allows the computation of the whole evolution curve over the entire loading history. We show that the SBEN variational formulation yields to a constrained time–space minimization problem. The cost function depends on stress, displacements, and plastic multiplier fields, which, naturally, leads to a mixed finite element discretization. The details of the solution algorithm are illustrated through the numerical study of the elastoplastic response of the thin and thick pressurized tubes, including inertia effects. Moreover, the balance of momentum is handled using two approaches. Firstly, the Schaefer superposition technique is considered, where the balance of momentum is exactly satisfied. Secondly, the balance of momentum is considered as an optimization constraint and satisfied only at the Gauss integration point level of the finite element problem. The accuracy and efficiency of the SBEN principle are assessed by comparing the numerical results with the analytical solutions (for the thin tube) and the predictions derived by the conventional incremental finite element procedure.

Acoustic scattering in nonhomogeneous media and the problem of discontinuous gradients: Analysis and inf‐sup stability in the method of finite spheres

AbstractIn this paper we focus on a meshfree formulation for the solution of time‐harmonic acoustic scattering problems and verify the stability of the procedure. The sound waves propagate in nonhomogeneous media, giving rise to discontinuities in the gradients of the pressure field across the interfaces between regions of different material properties. Meshfree methods usually do not reproduce accurately the discontinuities in a numerical solution. We overcome this issue by introducing Lagrange multiplier fields defined at the interfaces in order to treat the discontinuities in the gradients of the pressure field. The method does not depend on any kind of adjustable parameter. We show by a numerical study of the applicable inf‐sup conditions that the resulting mixed formulation leads to well‐posed problems. The use of the proposed method is illustrated in the solution of a number of problems of wave propagation through nonhomogeneous media.

On restricting first order form of gauge theories to one-loop order

The first order form of the Yang–Mills and Einstein–Hilbert actions are quantized, and it is shown how Green’s functions computed using the first and the second order form of these theories are related. Next we show how by use of Lagrange multiplier fields (LM), radiative effects beyond one-loop order can be eliminated. This allows one to compute Green’s functions exactly without loss of unitarity. The consequences of this restriction on radiative effects are examined for the Yang–Mills and Einstein–Hilbert actions. In these two gauge theories, we find that the quantized theory is both renormalizable and unitary once the LM field is used to eliminate effects beyond one-loop order.

The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

When the method of finite spheres is used for the solution of time-harmonic acoustic wave propagation problems in nonhomogeneous media, a mixed (or saddle-point) formulation is obtained in which the unknowns are the pressure fields and the Lagrange multiplier fields defined at the interfaces between the regions with distinct material properties. Then certain inf-sup conditions must be satisfied by the discretized spaces in order for the finite-dimensional problems to be well-posed. We discuss in this paper the analysis and use of these conditions. Since the conditions involve norms of functionals in fractional Sobolev spaces, we derive ‘stronger’ conditions that are simpler in form. These new conditions pave the way for the inf-sup testing, a tool for assessing the stability of the discretized problems.

On restricting to one-loop order the radiative effects in quantum gravity

The dimensionful nature of the coupling in the Einstein–Hilbert action in four dimensions implies that the theory is non-renormalizable; explicit calculation shows that beginning at two loop order, divergences arise that cannot be removed by renormalization without introducing new terms in the classical action. It has been shown that, by use of a Lagrange multiplier field to ensure that the classical equation of motion is satisfied in the path integral, radiative effects can be restricted to one-loop order. We show that by use of such Lagrange multiplier fields, the Einstein–Hilbert action can be quantized without the occurrence of non-renormalizable divergences. We then apply this mechanism to a model in which there is in addition to the Einstein–Hilbert action, a fully covariant action for a self-interacting scalar field coupled to the metric. It proves possible to restrict loop diagrams involving internal lines involving the metric to one-loop order; diagrams in which the scalar field propagates occur at arbitrary high order in the loop expansion. This model also can be shown to be renormalizable. Incorporating spinor and vector fields in the same way as scalar fields is feasible, and so a fully covariant Standard Model with a dynamical metric field can also be shown to be renormalizable.

Use of Lagrange multiplier fields to eliminate multiloop corrections

The problem of eliminating divergences arising in quantum gravity is generally addressed by modifying the classical Einstein-Hilbert action. These modifications might involve the introduction of local supersymmetry, the addition of terms that are higher-order in the curvature to the action, or invoking compactification of superstring theory from ten to four dimensions. An alternative to these approaches is to introduce a Lagrange multiplier field that restricts the path integral to field configurations that satisfy the classical equations of motion; this has the effect of doubling the usual one-loop contributions and of eliminating all effects beyond one loop. We show how this reduction of loop contributions occurs and find the gauge invariances present when such a Lagrange multiplier is introduced into the Yang-Mills and Einstein-Hilbert actions. Moreover, we quantize using the path integral, discuss the renormalization, and then show how Becchi-Rouet-Stora-Tyutin (BRST) invariance can be used to both demonstrate that unitarity is retained and to find BRST relations between Greens functions. In the Appendices, we show how the background field quantization can be implemented, consider the use of a Lagrange multiplier field to restrict higher-order contributions in supersymmetric theories, and derive the BRST equations satisfied by the generating functional.

A dual domain decomposition algorithm for the analysis of non-conforming isogeometric Kirchhoff–Love shells

Originally, Isogeometric Analysis is aimed at using geometric models for the structural analysis. The actual realization of this objective to complex real-world structures requires a special treatment of the non-conformities between the patches generated during the geometric modeling. Different advanced numerical tools now enable to analyze elaborated multipatch models, especially regarding the imposition of the interface coupling conditions. However, in order to push forward the isogeometric concept, a closer look at the algorithm of resolution for multipatch geometries seems crucial. Hence, we present a dual Domain Decomposition algorithm for accurately analyzing non-conforming multipatch Kirchhoff–Love shells. The starting point is the use of a Mortar method for imposing the coupling conditions between the shells. The additional degrees of freedom coming from the Lagrange multiplier field enable to formulate an interface problem, known as the one-level FETI problem. The interface problem is solved using an iterative solver where, at each iteration, only local quantities defined at the patch level (i.e. per sub-domain) are involved which makes the overall algorithm naturally parallelizable. We study the preconditioning step in order to get an algorithm which is numerically scalable. Several examples ranging from simple benchmark cases to semi-industrial problems highlight the great potential of the method.

Storm Catalog‐Based Analysis of Rainfall Heterogeneity and Frequency in a Complex Terrain

AbstractUrban development, topographic relief, and coastal boundaries can all exert influences on storm hydroclimatology, making rainfall and flood frequency analysis a major challenge. This study explores heterogeneity in extreme rainfall in the Baltimore Metropolitan region at small spatial scales using hydrometeorological analyses of major storm events in combination with hydroclimatological analyses based on storm catalogs developed using a 16‐year record of high‐resolution bias‐corrected radar rainfall fields. Our analyses demonstrate the potential for rainfall frequency methods using storm catalogs combined with stochastic storm transposition (SST); procedures are implemented for Dead Run, a small (14.3 km2) urban watershed located within the Baltimore Metropolitan area. The results point to the pronounced impact of complex terrain (including the Chesapeake Bay to the east, mountainous terrain to the west and urbanization in the region) on the regional rainfall climatology. Warm‐season thunderstorm systems are shown to be the dominant mechanism for generating extreme, short‐duration rainfall that leads to flash flooding. The SST approach is extended through the implementation of a multiplier field that accounts for spatial heterogeneities in extreme rainfall magnitude. SST‐based analyses demonstrate the need to consider rainfall heterogeneity at multiple scales when estimating the rainfall intensity‐duration‐frequency relationships.