Minimum theorems in 3D incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.     

An Integrated Procedure for the Structural Design of a Composite Rotor-Hydrofoil of a Water Current Turbine (WCT)

This paper shows an integrated structural design optimization of a composite rotor-hydrofoil of a water current turbine by means the finite elements method (FEM), using a Serial/Parallel mixing theory (Rastellini et al. Comput. Struct. 86:879–896, 2008, Martinez et al., 2007, Martinez and Oller Arch. Comput. Methods. 16(4):357–397, 2009, Martinez et al. Compos. Part B Eng. 42(2011):134–144, 2010) coupled with a fluid-dynamic formulation and multi-objective optimization algorithm (Gen and Cheng 1997, Lee et al. Compos. Struct. 99:181–192, 2013, Lee et al. Compos. Struct. 94(3):1087–1096, 2012). The composite hydrofoil of the turbine rotor has been design using a reinforced laminate composites, taking into account the optimization of the carbon fiber orientation to obtain the maximum strength and lower rotational-inertia. Also, these results have been compared with a steel hydrofoil remarking the different performance on both structures. The mechanical and geometrical parameters involved in the design of this fiber-reinforced composite material are the fiber orientation, number of layers, stacking sequence and laminate thickness. Water pressure in the rotor of the turbine is obtained from a coupled fluid-dynamic simulation (CFD), whose detail can be found in the reference Oller et al. (2012). The main purpose of this paper is to achieve a very low inertia rotor minimizing the start-stop effect, because it is applied in axial water flow turbine currently in design by the authors, in which is important to take the maximum advantage of the kinetic energy. The FEM simulation codes are engineered by CIMNE (International Center for Numerical Method in Engineering, Barcelona, Spain), COMPack for the solids problem application, KRATOS for fluid dynamic application and RMOP for the structural optimization. To validate the procedure here presented, many turbine rotors made of composite materials are analyzed and three of them are compared with the steel one.     

A comparative study between two smoothing strategies for the simulation of contact with large sliding

The numerical simulation of contact problems is still a delicate matter especially when large transformations are involved. In that case, relative large slidings can occur between contact surfaces and the discretization error induced by usual finite elements may not be satisfactory. In particular, usual elements lead to a facetization of the contact surface, meaning an unavoidable discontinuity of the normal vector to this surface. Uncertainty over the precision of the results, irregularity of the displacement of the contact nodes and even numerical oscillations of contact reaction force may result of such discontinuity. Among the existing methods for tackling such issue, one may consider mortar elements (Fischer and Wriggers, Comput Methods Appl Mech Eng 195:5020–5036, 2006; McDevitt and Laursen, Int J Numer Methods Eng 48:1525–1547, 2000; Puso and Laursen, Comput Methods Appl Mech Eng 93:601–629, 2004), smoothing of the contact surfaces with additional geometrical entity (B-splines or NURBS) (Belytschko et al., Int J Numer Methods Eng 55:101–125, 2002; Kikuchi, Penalty/finite element approximations of a class of unilateral contact problems. Penalty method and finite element method, ASME, New York, 1982; Legrand, Modèles de prediction de l’interaction rotor/stator dans un moteur d’avion Thèse de doctorat. PhD thesis, École Centrale de Nantes, Nantes, 2005; Muñoz, Comput Methods Appl Mech Eng 197:979–993, 2008; Wriggers and Krstulovic-Opara, J Appl Math Mech (ZAMM) 80:77–80, 2000) and, the use of isogeometric analysis (Temizer et al., Comput Methods Appl Mech Eng 200:1100–1112, 2011; Hughes et al., Comput Methods Appl Mech Eng 194:4135–4195, 2005; de Lorenzis et al., Int J Numer Meth Eng, in press, 2011). In the present paper, we focus on these last two methods which are combined with a finite element code using the bi-potential method for contact management (Feng et al., Comput Mech 36:375–383, 2005). A comparative study focusing on the pros and cons of each method regarding geometrical precision and numerical stability for contact solution is proposed. The scope of this study is limited to 2D contact problems for which we consider several types of finite elements. Test cases are given in order to illustrate this comparative study.     

Twenty Years of Magnon Bose Condensation and Spin Current Superfluidity in 3He-B

20 years ago a new quantum state of matter was discovered and identified (Borovik-Romanov et al. in JETP Lett. 40:1033, 1984; 45:124, 1987; 47:478, 1988; Fomin in JETP Lett. 40:1037, 1984; Borovik-Romanov et al. in Sov. Phys. JETP 61:1199, 1985; Fomin in Sov. Phys. JETP 61:1207, 1985; Bunkov et al. in JETP Lett. 43:168, 1986). The observed dynamic quantum state of spin precession in superfluid ³He-B bears the properties of spin current superfluidity, Bose condensation of spin waves—magnons, off-diagonal long-range order and related phenomena of quantum coherence.     

Quantized Vortex State in hcp Solid 4He

The quantized vortex state appearing in the recently discovered new states in hcp ⁴He since their discovery (Kim and Chan, Nature, 427:225–227, 2004; Science, 305:1941, 2004) is discussed. Special attention is given to evidence for the vortex state as the vortex fluid (VF) state (Anderson, Nat. Phys., 3:160–162, 2007; Phys. Rev. Lett., 100:215301, 2008; Penzev et al., Phys. Rev. Lett., 101:065301, 2008; Nemirovskii et al., arXiv:0907.0330, 2009) and its transition into the supersolid (SS) state (Shimizu et al., arXiv:0903.1326, 2009; Kubota et al., J. Low Temp. Phys., 158:572–577, 2010; J. Low Temp. Phys., 162:483–491, 2011). Its features are described. The historical explanations (Reatto and Chester, Phys. Rev., 155(1):88–100, 1967; Chester, Phys. Rev. A, 2(1):256–258, 1970; Andreev and Lifshitz, JETP Lett., 29:1107–1113, 1969; Leggett, Phys. Rev. Lett., 25(22), 1543–1546, 1970; Matsuda and Tsuneto, Prog. Theor. Phys., 46:411–436, 1970) for the SS state in quantum solids such as solid ⁴He were based on the idea of Bose Einstein Condensation (BEC) of the imperfections such as vacancies, interstitials and other possible excitations in the quantum solids which are expected because of the large zero-point motions. The SS state was proposed as a new state of matter in which real space ordering of the lattice structure of the solid coexists with the momentum space ordering of superfluidity. A new type of superconductors, since the discovery of the cuprate high T _c superconductors, HTSCs (Bednorz and Mueller, Z. Phys., 64:189, 1986), has been shown to share a feature with the vortex state, involving the VF and vortex solid states. The high T _c s of these materials are being discussed in connection to the large fluctuations associated with some other phase transitions like the antiferromagnetic transition in addition to that of the low dimensionality. The supersolidity in the hcp solid ⁴He, in contrast to the new superconductors which have multiple degrees of freedom of the Cooper pairs with spin as well as angular momentum freedom, has a unique feature of possessing possibly only the momentum fluctuations and vortex ring excitations associated with the possible low dimensional fluctuations of the subsystem(s). The high onset temperature of the VF state can be understood by considering thermally excited low D quantized vortices and it may be necessary to seek low dimensional sub-systems in hcp He which are hosts for vortices.     

Transverse deflection of a clamped mild-steel plate

The theory of plastic flow by extended slip (Acta Mech 223:655–668, 2012; Philos Mag A 91:3343–3357, 2011; Z Angew Math Mech 84:266–279, 2004; Q J Mech Appl Math 52:645–662, 1999) is applied to a problem of bi-axial strain: the transverse plastic deflection, by means of a flat-ended punch, of a clamped plate of mild-steel. Two new theorems concerned with the Mechanics of Plates are presented. It is shown that, if the static shear yield stress of the plate material remains independent of strain, then the load–deflection relation for the punch, in the case of quasi-static punching of a plate clamped along a closed arbitrary contour, obeys an exactly linear theoretical relation. This prediction is then confirmed by experiments carried out at quasi-static rates of loading with thin plates of hot-rolled mild-steel. It is demonstrated by experiment, in the case of concentric circular punch and clamp contours, that the load–deflection relation for the punch remains linear to within <1 % provided that the maximum principal strain within the free domain of the plate does not exceed the yield-point elongation strain of the mild-steel concerned.     

A three-dimensional model for flow pumping in a microchannel inspired by insect respiration

We present a three-dimensional model for flow pumping in a channel induced by two moving contractions from the upper wall. This pumping model is inspired by insect respiration processes, specifically, the rhythmic collapses that take place within their tracheal tube networks. The present work is a natural extension of our previous theoretical and numerical investigations of a two-dimensional insect-inspired micropumping model, which accounts for three-dimensional effects and further validates our insect-inspired pumping paradigm (Aboelkassem and Staples in Acta Mech 223(3):463–480, 2012a; Theor Comput Fluid Dyn, 2012b. doi:10.1007/s00162-012-0269-7). The formal goal of this article is to compare three-dimensional Stokeslets-meshfree numerical results with results from our previous two-dimensional analytical pumping model. We use regularized Stokeslets-meshfree computations in three dimensions to reconstruct the flow motions induced by wall contractions and to calculate the time-averaged net flow pumping rate. The results show that, although the net flow rate distribution as a function of the wall motion time (phase) lag parameter for the three-dimensional Stokeslets-meshfree computations and the two-dimensional analytical model displays some differences, the same basic features appear in both cases, leading to the same general conclusions about the proposed pumping paradigm.     

Constraints on the applicability range of Hill’s criterion: strong orthotropy or transverse isotropy

Commonly used orthotropic Hill’s criterion of plastic flow initiation (Hill in Proc R Soc Lond A 193:281–297, 1948) suffers from some constraints and inconsistencies, which are of two different origins. Firstly, in case of high orthotropy degree, the quadratic form corresponding to Hill’s criterion may change type from convex and closed elliptic to concave and open hyperbolic in the deviatoric stress space (Ottosen and Ristinmaa in the mechanics of constitutive modeling, Elsevier, Amsterdam, 2005). Secondly, application of classical Hill’s criterion to transversely isotropic materials shows a discrepancy between Hill’s limit curves in the transverse isotropy plane and the Huber-von Mises prediction for isotropic materials (Huber 1904; von Mises 1913). The basic result of the present paper is to propose the new transversely isotropic von Mises–Hu–Marin’s-type criterion of hexagonal symmetry that is free from both constraints. The new enhanced Hu–Marin’s-type limit surface represents an elliptic cylinder, the axis of which is proportional to stress/strength, in contrast to Hill’s-type limit surface possessing the hydrostatic axis. Hence, this condition does not exhibit the deviatoricity property, which is a price for coincidence with the Huber–von Mises condition in the transverse isotropy plane, but with cylindricity ensured for an arbitrarily high orthotropy degree. The hybrid-type transversely isotropic Hu–Marin’s criterion of mixed symmetry based on additional biaxial bulge test, capable of fitting experimental findings for some complex composites, is also proposed. Application of this criterion has been verified for a unidirectional SiC/Ti composite examined by Herakovich (Thermal stresses V, Lastran Corp. Publ. Division, pp 1–142, 1999).     

Thermodynamics and thermal decomposition for shape memory effects with crystallization based on dissipation and logarithmic strain

The present effort provides a 3-D thermodynamic framework generalizing the 1-D modeling of 2-way shape memory materials described by Westbrook et al. (J. Eng. Mater. Technol. 312:041010, 2010) and Chung et al. (Macromolecules 41:184–192, 2008), while extending the strain-induced crystallization and shape memory approaches of Rao and Rajagopal (Interfaces Free Bound. 2:73–94, 2000; Int. J. Solids Struct. 38:1149–1167, 2001), Barot and Rao (Z. Angew. Math. Phys. 57:652–681, 2006), and Barot et al. (Int. J. Eng. Sci. 46:325–351, 2008) to include finite thermal expansion within a logarithmic strain basis. The free energy of newly-formed orthotropic crystallites is assumed additive, with no strains in their respective configurations of formation. A multiplicative decomposition is assumed for the assumed thermoelastic orthotropic expansional strains of the respective crystallites. The properties of the crystallites are allowed to depend both on current temperature and their respective temperatures of formation. The entropy production rate relation is written in the frame rotating with the logarithmic spin and produces stress and entropy relations incorporating the integrated configurational free energies, and a driving term for the crystallization analogous to that obtained by the previous studies of Rao et al. The salient attributes of the 1-D modeling of Westbrook et al. are recovered, and applications are discussed.     

An extension of the noncommutative Bergman’s ring with a large number of noninvertible elements

For a prime number \(p\) , Bergman (Israel J Math 18:257–277, 1974) established that \(\mathrm {End}(\mathbb {Z}_{p} \times \mathbb {Z}_{p^{2}})\) is a semilocal ring with \(p^{5}\) elements that cannot be embedded in matrices over any commutative ring. In an earlier paper Climent et al. (Appl Algebra Eng Commun Comput 22(2):91–108, 2011), the authors presented an efficient implementation of this ring, and introduced a key exchange protocol based on it. This protocol was cryptanalyzed by Kamal and Youssef (Appl Algebra Eng Commun Comput 23(3–4):143–149, 2012) using the invertibility of most elements in this ring. In this paper we introduce an extension of Bergman’s ring, in which only a negligible fraction of elements are invertible, and propose to consider a key exchange protocol over this ring.     

Finite and spectral cell method for wave propagation in heterogeneous materials

In the current paper we present a fast, reliable technique for simulating wave propagation in complex structures made of heterogeneous materials. The proposed approach, the spectral cell method, is a combination of the finite cell method and the spectral element method that significantly lowers preprocessing and computational expenditure. The spectral cell method takes advantage of explicit time-integration schemes coupled with a diagonal mass matrix to reduce the time spent on solving the equation system. By employing a fictitious domain approach, this method also helps to eliminate some of the difficulties associated with mesh generation. Besides introducing a proper, specific mass lumping technique, we also study the performance of the low-order and high-order versions of this approach based on several numerical examples. Our results show that the high-order version of the spectral cell method together requires less memory storage and less CPU time than other possible versions, when combined simultaneously with explicit time-integration algorithms. Moreover, as the implementation of the proposed method in available finite element programs is straightforward, these properties turn the method into a viable tool for practical applications such as structural health monitoring [1–3], quantitative ultrasound applications [4], or the active control of vibrations and noise [5, 6].     

Simulation of concrete flow in V-funnel test and the proper range of viscosity and yield stress for SCC

The present paper highlights the flow simulation of self consolidating concrete (SCC) in V-funnel test that is used to determine the concrete filling ability and its resistance against segregation. Simulations were performed using a two-dimensional smoothed particle hydrodynamic (SPH) method to determine the discharge time where SCC was considered as a homogeneous Bingham fluid. The numerical predictions are lower than experimental data because of the assumptions of two-dimensional and homogeneous flow. Having the SPH method employed, SCCs with different viscosities and yield stresses were simulated to compare the discharge time with the suggested criteria in EFNARC (2002) and (2005) guidelines. Based on simulations results, the appropriate range of viscosities and yield stresses as well as a relation between rheological properties and discharge time for SCC taking into account EFNARC (2002) and (2005) guidelines are suggested. Using the suggested relations, one can assess the proper SCC filling ability without conducting the V-funnel test.     

Neumann exterior wave propagation problems: computational aspects of 3D energetic Galerkin BEM

In this work, we consider as model problem an exterior 3D wave propagation Neumann problem reformulated in terms of a space–time hypersingular boundary integral equation with retarded potentials. This latter is set in the so-called energetic weak form, recently proposed in Aimi et al. (Int J Numer Methods Eng 80:1196–1240, 2009; CMES 58:185–219, 2010), regularized as in Frangi (Int J Numer Methods Eng 45:721–740, 1999) and then approximated by the Galerkin boundary element method. Details on the discretization phase and, in particular, on the computation of integrals, double in time and double in space, constituting the elements of the final linear system matrix are given and analyzed. Various numerical results and simulations are presented and discussed.     

Numerical simulation of buoyant drops suspended in Poiseuille flow at nonzero Reynolds numbers

Suspensions of two-dimensional buoyant drops in Poiseuille flow are studied at nonzero Reynolds numbers by numerical simulations. The flow is studied as a function of the Froude number, the Reynolds number, the Capillary number and the density ratio. First, the lateral migration of a drop is studied. Results agree with two-dimensional simulations of solid circular cylinders by Feng et al. (J Fluid Mech 261:95–134, 1994; J Fluid Mech 277:271–301, 1994) qualitatively. At a relatively large Reynolds number (120) and a moderate Froude number (43), a drop shows oscillations across the channel and does not obtain a stable equilibrium position. Simulations are also performed at low and moderate area fractions (0.22, 0.44). It is found that the effective viscosity strongly depends on the Froude number for heavy drops (α > 1). The effective viscosity changes with the Froude number for light drops as well (α < 1), but to a lower extent. The distribution and the fluctuation energy of drops across the channel are non-uniform for buoyant drops that depend on the Froude number. The density ratio also affects the distribution and fluctuation energy of drops across the channel. The effect of the Reynolds number on the effective viscosity of the suspension is also investigated.     

Reply: On Role of Symmetries in Kelvin Wave Turbulence

In the Ref. (Lebedev and L’vov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0215-2), this issue, two of us (VVL and VSL) considered symmetry restriction on the interaction coefficients of Kelvin waves and demonstrated that linear in small wave vector asymptotic, obtained analytically, is not forbidden, as Kosik and Svistunov (KS) expect by naive reasoning. Here we discuss this problem in additional details and show that theoretical objections by KS, presented in Ref. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z), this issue, are irrelevant and their recent numerical simulation, presented in Ref. (Kozik and Svistunov in arXiv:1007.4927v1, 2010) is hardly convincing. There is neither proof of locality nor any refutation of the possibility of linear asymptotic of interaction vertices in the KS texts, Refs. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z; arXiv:1006.0506v1, 2010). Therefore we can state again that we have no reason to doubt in this asymptote, that results in the L’vov–Nazarenko energy spectrum of Kelvin waves.     

A note on cyclic codes from APN functions

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding (SIAM J Discret Math 27(4):1977–1994, 2013), Ding and Zhou (Discret Math, 2014) constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN function $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$ . From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty }$ defined by $s_t={\mathrm {Tr}}((1+\alpha ^t)^e)$ , where $\alpha $ is a primitive element in ${\mathrm {GF}}(q)$ . These techniques can also be applied to other open problems in Ding (SIAM J Discret Math 27(4):1977–1994, 2013), Ding and Zhou (Discret Math, 2014).     

Discovering a junction tree behind a Markov network by a greedy algorithm

In our paper (Annals of Operations Research, 193:71–90, 2012) we introduced a special kind of k?1-width junction tree, called k-th order cherry tree in order to approximate a joint probability distribution. The approximation is the best if the Kullback–Leibler divergence between the true joint probability distribution and the approximating one is minimal. Finding the best approximating k?1-width junction tree probability distribution is NP-complete if 2<k<d?1, where d is the dimension of the joint probability distribution (see Karger and Srebro in 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2001, Malvestuto in Kybernetika 48:825–844, 2012). In Broad Research in Artificial Intelligence and Neuroscience (BRAIN), special issue on complexity in sciences and artificial intelligence, pp. 40–45, 2010 we also proved that the best approximating k?1-width junction tree probability distribution can be embedded into a k-th order cherry tree probability distribution. We introduce here a greedy algorithm resulting very good approximations in reasonable computing time. We prove then that if the Markov network which encodes the conditional independences of the multivariate probability distribution fulfills some requirements then our greedy algorithm is able to find the true probability distribution. Our algorithm uses just the k-th order marginal probability distributions as input. We compare the results of the greedy algorithm proposed in this paper with the greedy algorithm proposed by Malvestuto (IEEE Trans. Syst. Man Cybern. 21:1287–1294, 1991).     

Experimental Investigation of Nucleate Boiling on a Thermal Capacitive Heater Under Variable Gravity Conditions

In the present work the behavior of single vapor bubbles of FC-72, generated on a thermal capacitive heater element, has been investigated during microgravity. A newly developed heater design allows temperature measurements by highspeed infrared thermography on the backside of the heater surface at a distance of approx. 800 nm from the fluid/heater-interface. The employed heater was manufactured by Physical Vapor Deposition (PVD) of a chromium based layer for better emissivity (Slomski et al., Mater Sci Technol 41:161?C165, 2010) and a pure chromium heating layer supplying the energy required for bubble generation and sustainment by electrical heating. The thermal diffusivity of the employed Calcium Flouride (CaF) heater substrate is comparable to the thermal diffusivity of stainless steel, which makes this heater design very close to technical applications. The acquired transient temperature fields of the heater surface allow numerical determination of the local heat flux from the heater surface to the fluid. A local temperature drop and high heat fluxes have been observed in the vicinity of the 3-phase contact line. This effect has already been reported by former publications for thin stainless steel foil heaters (Stephan and Hammer, Int J Heat Mass Transfer 30:119?C125, 1994; Wagner et al., Int J Heat Mass Transfer 42:875?C883, 2006) and is also confirmed for heaters with significantly higher thermal capacities.     

Theoretical Analysis of Neutron and X-ray Scattering Data on 3He

X-ray scattering experiments on bulk liquid ³He (Albergamo et al. in Phys. Rev. Lett. 99:205301, 2007; Schmets and Montfrooij in Phys. Rev. Lett. 100:239601, 2008; Albergamo et?al. in Phys. Rev. Lett. 100:239602, 2008) have indicated the possibility of the existence of a sharp collective mode at large momentum transfers. We address this issue within a manifestly microscopic theory of excitations in a Fermi fluid that can be understood as proper generalization of the time-honored theory of Jackson, Feenberg, and Campbell (Jackson in Phys. Rev. A 8:1529, 1973; Feenberg in Theory of Quantum Fluids, 1969; Chang and Campbell in Phys. Rev. B 13:3779, 1976) of excitations in ⁴He. We show that both neutron and X-ray data can be well explained within a theory where the high momentum excitations lie in fact inside the particle-hole continuum. ??Pair fluctuations?? contribute a sharpening of the mode compared to the random phase approximation (RPA). When the theoretical results are convoluted with the experimental resolution, the agreement between theory and X-ray data is quite good.