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Development of Ion Stopping Models for HED Plasmas Using Unified Self-Consistent Field Models and Self-Consistent Electron Distributions
We have implemented several corrections to the electronic stopping power model combining the RPA dielectric response formalism and local density approximation with electronic density distribution calculated in an average atom model. These modifications include strong collision correction, local field correction, electron binding energy correction, and the Barkas effect. The combined results bring the RPA-LDA stopping power in cold targets to closer agreements with experiments for a wide range of materials. The same method is then applied to the stopping of ions in warm dense plasmas. The computational framework developed during this project is publicly available on GitHub (https://github.com/dedx-erpa/dedx). Tabulated data for protons in cold target for common materials are located in the data/ subdirectory of the repository.
Formal Definitions and Performance Comparison of Consistency Models for Parallel File Systems
The semantics of HPC storage systems are defined by the consistency models to which they abide. Storage consistency models have been less studied than their counterparts in memory systems, with the exception of the POSIX standard and its strict consistency model. The use of POSIX consistency imposes a performance penalty that becomes more significant as the scale of parallel file systems increases and the access time to storage devices, such as node-local solid storage devices, decreases. While some efforts have been made to adopt relaxed storage consistency models, these models are often defined informally and ambiguously as by-products of a particular implementation. Here in this work, we establish a connection between memory consistency models and storage consistency models and revisit the key design choices of storage consistency models from a high-level perspective. Further, we propose a formal and unified framework for defining storage consistency models and a layered implementation that can be used to easily evaluate their relative performance for different I/O workloads. Finally, we conduct a comprehensive performance comparison of two relaxed consistency models on a range of commonly seen parallel I/O workloads, such as checkpoint/restart of scientific applications and random reads of deep learning applications. We demonstrate that for certain I/O scenarios, a weaker consistency model can significantly improve the I/O performance. For instance, in small random reads that are typically found in deep learning applications, session consistency achieved a 5x improvement in I/O bandwidth compared to commit consistency, even at small scales.
A consistent and conservative Phase-Field method for multiphase incompressible flows
In the present study, a consistent and conservative Phase-Field method, including both the model and scheme, is developed for multiphase flows with an arbitrary number of immiscible and incompressible fluid phases. The consistency of mass conservation and the consistency of mass and momentum transport are implemented to address the issue of physically coupling the Phase-Field equation, which locates different phases, to the hydrodynamics. These two consistency conditions, as illustrated, provide the “optimal” coupling because (i) the new momentum equation resulting from them is Galilean invariant and implies the kinetic energy conservation, regardless of the details of the Phase-Field equation, and (ii) failures of satisfying the second law of thermodynamics or the consistency of reduction of the multiphase flow model only result from the same failures of the Phase-Field equation but are not due to the new momentum equation. Physical interpretation of the consistency conditions and their formulations are first provided, and general formulations that are obtained from the consistency conditions and independent of the interpretation of the velocity are summarized. Then, the present consistent and conservative multiphase flow model is completed by selecting a reduction consistent Phase-Field equation. Several novel techniques are developed to inherit the physical properties of the multiphase flows after discretization, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, and the general theorems to preserve the consistency conditions on the discrete level. Equipped with those novel techniques, a consistent and conservative scheme for the present multiphase flow model is developed and analyzed. The scheme satisfies the consistency conditions, conserves the mass and momentum, and assures the summation of the volume fractions to be unity, on the fully discrete level and for an arbitrary number of phases. All those properties are numerically validated. Finally, numerical applications demonstrate that the present model and scheme are robust and effective in studying complicated multiphase dynamics, especially for those with large-density ratios.
A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models
In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the consistency of reduction. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the consistency of reduction is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the consistency of mass conservation and the consistency of mass and momentum transport, thanks to the consistent formulation. Finally, it is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio.
Interface Consistency: Phase I Results & Phase II Status
Future exploration missions will rely on designing and developing vehicles and complex systems from within NASA and through multiple external commercial partners to meet mission goals. Despite existing consistency-related agency requirements, NASA’s approach to commercial spaceflight development encourages providers’ flexibility and innovation. This strategy is resulting in significant design diversity across Artemis vehicles. Design best practices and guidelines champion interface consistency to promote mental model development and knowledge transfer. However, research investigating the benefits of consistency is mixed, and little is known about its role in complex systems. Determining the level of risk that system diversity presents is difficult, as there is no established method for quantifying the degree of consistency within and across interfaces, nor is there information about the differential impacts of different types of inconsistency. Phase I of this project (Characterization and Measurement) served as a starting point to better understand the construct of consistency, its application, and the range of studies and methods for measuring it. The project team created a taxonomy of consistency to apply to interfaces as a framework to guide the development of tools to assess intersystem consistency. Checklist and cognitive walkthrough methods were developed for use by human factors (HF) and human-computer interaction (HCI) experts. The Intersystem Consistency Scale (ICS) was developed for interface evaluations with crew. A pilot study evaluated the methods’ ability to distinguish differences between Artemis-like prototype pairs exhibiting either high or low design consistency. In addition, click errors and time on task were collected within the ICS (crew-like) group. Results from our exploratory analysis and lessons learned from the pilot study will be discussed. The project team will also present the status of Phase II (Risk Assessment, Standards and Guidelines). This includes incorporating feedback to redesign the assessment tools, and inputs from displays and training Subject Matter Experts to update tasks and prototype designs. The team will present the risk assessment study design to identify the types and levels of inconsistency that pose the greatest risk to performance. Plans to apply these results toward agency standards and guideline recommendations will also be discussed.
Objective past of a quantum universe: Redundant records of consistent histories
Motivated by the advances of quantum Darwinism and recognizing the role played by redundancy in identifying the small subset of quantum states with resilience characteristic of objective classical reality, we explore the implications of redundant records for consistent histories. The consistent histories formalism is a tool for describing sequences of events taking place in an evolving closed quantum system. A set of histories is consistent when one can reason about them using Boolean logic, i.e., when probabilities of sequences of events that define histories are additive. However, the vast majority of the sets of histories that are merely consistent are flagrantly nonclassical in other respects. This embarras de richesses (known as the set selection problem) suggests that one must go beyond consistency to identify how the classical past arises in our quantum universe. The key intuition we follow is that the records of events that define the familiar objective past are inscribed in many distinct systems, e.g., subsystems of the environment, and are accessible locally in space and time to observers. We identify histories that are not just consistent but redundantly consistent using the partial-trace condition introduced by Finkelstein as a bridge between histories and decoherence. The existence of redundant records is a sufficient condition for redundant consistency. It selects, from the multitude of the alternative sets of consistent histories, a small subset endowed with redundant records characteristic of the objective classical past. Furthermore, the information about an objective history of the past is then simultaneously within reach of many, who can independently reconstruct it and arrive at compatible conclusions in the present.
Full versus quasiparticle self-consistency in vertex-corrected GW approaches
Here, using seven semiconductors/insulators with band gaps covering the range from 1 eV to 10 eV we systematically explore the performance of two different variants of self-consistency associated with famous Hedin's system of equations: the full self-consistency and the so called quasi-particle approximation to it. The pros and cons of these two variants of self-consistency are sufficiently well documented in literature for the simplest GW approximation to the Hedin's equations. Our study, therefore, aims primarily at the level of theory beyond GW approximation, i.e. at the level of theory which includes vertex corrections. Whereas quasi-particle self-consistency has certain advantages at GW level (well known fact), the situation becomes quite different when vertex corrections are included. In the variant with full self-consistency, vertex corrections (both for polarizability and for self energy) systematically reduce the calculated band gaps making them closer to the experimental values. In the variant with quasi-particle self-consistency, however, an inclusion of the same diagrams has considerably larger effect and calculated band gaps become severely underestimated. Different effect of vertex corrections in two variants of self-consistency can be related to the Z-factor cancellation which plays positive role in quasi-particle self-consistency at GW level of theory but appears to be destructive for the quasi-particle approximation when higher order diagrams are included. Second result of our study is that we were able to reproduce the results obtained with the Questaal code using our FlapwMBPT code when the same variant of self-consistency (quasi-particle) and the same level of vertex corrections (for polarizability only, static approximation for screened interaction, and Tamm-Dancoff approximation for the Bethe-Salpeter equation) are used.
The different varieties of the Suyama-Yamaguchi consistency relation and its violation as a signal of statistical inhomogeneity
We present the different consistency relations that can be seen as variations of the well known Suyama-Yamaguchi (SY) consistency relation τ{sub NL}≥((6/5)f{sub NL}){sup 2}, the latter involving the levels of non-gaussianity f{sub NL} and τ{sub NL} in the primordial curvature perturbation ζ. It has been (implicitly) claimed that the following variation: τ{sub NL}(k{sub 1},k{sub 3})≥((6/5)){sup 2}f{sub NL}(k{sub 1})f{sub NL}(k{sub 3}), which we call ''the fourth variety'', in the collapsed (for τ{sub NL}) and squeezed (for f{sub NL}) limits is always satisfied independently of any physics; however, the proof depends sensitively on the assumption of scale-invariance (expressing this way the fourth variety of the SY consistency relation as τ{sub NL}≥((6/5)f{sub NL}){sup 2}) which only applies for cosmological models involving Lorentz-invariant scalar fields (at least at tree level), leaving room for a strong violation of this variety of the consistency relation when non-trivial degrees of freedom, for instance vector fields, are in charge of the generation of the primordial curvature perturbation. With this in mind as a motivation, we explicitly state, in the first part of this work, under which conditions the SY consistency relation has been claimed to hold in its different varieties (implicitly) presented in the literature since its inception back in 2008; as a result, we show for the first time that the variety τ{sub NL}(k{sub 1},k{sub 1})≥((6/5)f{sub NL}(k{sub 1})){sup 2}, which we call ''the fifth variety'', is always satisfied even when there is strong scale-dependence and high levels of statistical anisotropy as long as statistical homogeneity holds: thus, an observed violation of this specific variety would prevent the comparison between theory and observation, shaking this way the foundations of cosmology as a science. In the second part, we concern about the existence of non-trivial degrees of freedom, concretely vector fields for which the levels of non-gaussianity have been calculated for very few models; among them, and by making use of the δN formalism at tree level, we study a class of models that includes the vector curvaton scenario, vector inflation, and the hybrid inflation with coupled vector and scalar ''waterfall field'' where ζ is generated at the end of inflation, finding that the fourth variety of the SY consistency relation is indeed strongly violated for some specific wavevector configurations while the fifth variety continues to be well satisfied. Finally, as a byproduct of our investigation, we draw attention to a quite recently demonstrated variety of the SY consistency relation: τ{sup iso}{sub NL}≥((6/5)f{sup iso}{sub NL}){sup 2}, in scenarios where scalar and vector fields contribute to the generation of the primordial curvature perturbation; this variety of the SY consistency relation is satisfied although the isotropic pieces of the non-gaussianity parameters receive contributions from the vector fields. We discuss further implications for observational cosmology.
SU-E-T-676: Reproducibility and Consistency of Two SunNuclear 3D Scanning Tanks
Purpose: To determine if two Sun Nuclear 3D Scanning (SNC 3DS) tanks collect reproducible and consistent data and test the precision of the SNC Dosimetry auto-setup. Methods: Percent depth doses (PDDs) and profiles were collected on two SNC 3DS tanks with a Varian TrueBeam linear accelerator. SNC Dosimetry auto-setup application was used with CC13 ionization chambers. After auto-setup, collimator light field was checked manually against the position of the chamber. Comparing measured data for repeated measurements with tank 1 allowed evaluation of SNC-3DS auto-setup and tank reproducibility. Comparing measured data between tanks 1 and 2 allowed evaluation of consistency between tanks. Results: Preliminary results showed reproducibility of depth of maximum dose (Dmax) of 0.38mm for a 10cmx10cm field and 0.67mm for 30cmx30cm on a single tank. PDD values at 5cm, 10cm, and 20cm depths were reproducible within 0.26%. Consistency of Dmax between tanks was 0.17mm for a 10cmx10cm field and 0.44mm for 30cmx30cm. PDD values at 5cm, 10cm, and 20cmwere consistent within 0.06%. Profiles showed reproducibility in field width within 0.4mm for a 10cmx10cm field and 0.7mm for a 30cmx30cm field. Profiles showed consistency in field width within 0.2mm for 10cmx10cm and 30cmx30cm field sizes. Penumbra width was reproducible and consistent to under 0.5mm except for 30cmx30cm field size at 30cm depth where the reproducibility was 2.2mm and the consistency was 2.6mm. Conclusion: In conclusion, the SNC 3DS tank shows good reproducibility in measured data. Since the tank to tank variation in measured data is within the uncertainty of repeated single tank measurements the tanks also perform consistently.
Injection of a Self-Consistent Beam at the Spallation Neutron Source
We plan to demonstrate the injection of a self-consistent beam into the Spallation Neutron Source (SNS). Self-consistent beams are defined to be ellipsoidal distributions with uniform density and to retain these properties under all linear transformations. Self-consistent distributions may generate very little halo if realized in practice. Some may also be manipulated to generate flat beams. Self-consistent distributions involve very special relationships between the phase space coordinates, making them difficult to realize experimentally. One self-consistent distribution, the 2D rotating distribution, can be painted into the SNS ring, with slight modification of the lattice. However, it is unknown how robust self-consistent distributions will be under real world transport in the presence of nonlinearities and other collective effects. This paper studies these issues and the mitigation of unwanted effects by applying realistic detailed computational models to the simulation of the injection of rotating beams into SNS. The result is a feasible prescription for the injection of a rotating self-consistent distribution into the SNS ring.
VerifyIO: Ensuring Correctness of Consistency Semantics in Parallel I/O
Abstract—High-performance computing (HPC) applications generate and consume substantial amounts of data, typically managed by parallel file systems. These applications access file systems either through the POSIX interface or by using highlevel I/O libraries. While the POSIX consistency model remains dominant in HPC, emerging file systems and popular I/O libraries increasingly adopt alternative consistency models that relax semantics in various ways, creating significant challenges for correctness and portability. This paper addresses these challenges by proposing a trace-driven I/O consistency verification workflow, implemented in our open-source tool, VerifyIO, which collects execution traces, detects data conflicts, and verifies proper synchronization against specified consistency models. Our extensive evaluation of 91 test case executions across three widely used I/O libraries with four I/O consistency models reveals critical consistency issues at both application and implementation levels.
Consistent searches for SMEFT effects in non-resonant dijet events
Here, we investigate the bounds which can be placed on generic new-physics contributions to dijet production at the LHC using the framework of the Standard Model Effective Field Theory, deriving the first consistently-treated EFT bounds from non-resonant high-energy data. We recast an analysis searching for quark compositeness, equivalent to treating the SM with one higher-dimensional operator as a complete UV model. In order to reach consistent, model-independent EFT conclusions, it is necessary to truncate the EFT effects consistently at order $1/\Lambda^2$ and to include the possibility of multiple operators simultaneously contributing to the observables, neither of which has been done in previous searches of this nature. Furthermore, it is important to give consistent error estimates for the theoretical predictions of the signal model, particularly in the region of phase space where the probed energy is approaching the cutoff scale of the EFT. There are two linear combinations of operators which contribute to dijet production in the SMEFT with distinct angular behavior; we identify those linear combinations and determine the ability of LHC searches to constrain them simultaneously. Consistently treating the EFT generically leads to weakened bounds on new-physics parameters. These constraints will be a useful input to future global analyses in the SMEFT framework, and the techniques used here to consistently search for EFT effects are directly applicable to other off-resonance signals.
Self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation theorem and the exact-exchange kernel
A self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem, employing the frequency-dependent exact exchange kernel f{sub x} is presented. The resulting SC-exact-exchange-only (EXX)-ACFD method leads to even more accurate correlation potentials than those obtained within the direct random phase approximation (dRPA). In contrast to dRPA methods, not only the Coulomb kernel but also the exact exchange kernel f{sub x} is taken into account in the EXX-ACFD correlation which results in a method that, unlike dRPA methods, is free of self-correlations, i.e., a method that treats exactly all one-electron systems, like, e.g., the hydrogen atom. The self-consistent evaluation of EXX-ACFD total energies improves the accuracy compared to EXX-ACFD total energies evaluated non-self-consistently with EXX or dRPA orbitals and eigenvalues. Reaction energies of a set of small molecules, for which highly accurate experimental reference data are available, are calculated and compared to quantum chemistry methods like Møller-Plesset perturbation theory of second order (MP2) or coupled cluster methods [CCSD, coupled cluster singles, doubles, and perturbative triples (CCSD(T))]. Moreover, we compare our methods to other ACFD variants like dRPA combined with perturbative corrections such as the second order screened exchange corrections or a renormalized singles correction. Similarly, the performance of our EXX-ACFD methods is investigated for the non-covalently bonded dimers of the S22 reference set and for potential energy curves of noble gas, water, and benzene dimers. The computational effort of the SC-EXX-ACFD method exhibits the same scaling of N{sup 5} with respect to the system size N as the non-self-consistent evaluation of only the EXX-ACFD correlation energy; however, the prefactor increases significantly. Reaction energies from the SC-EXX-ACFD method deviate quite little from EXX-ACFD energies obtained non-self-consistently with dRPA orbitals and eigenvalues, and the deviation reduces even further if the Coulomb kernel is scaled by a factor of 0.75 in the dRPA to reduce self-correlations in the dRPA correlation potential. For larger systems, such a non-self-consistent EXX-ACFD method is a competitive alternative to high-level wave-function-based methods, yielding higher accuracy than MP2 and CCSD methods while exhibiting a better scaling of the computational effort than CCSD or CCSD(T) methods. Moreover, EXX-ACFD methods were shown to be applicable in situation characterized by static correlation.
Fully consistent Eulerian Monte Carlo fields method for solving probability density function transport equations in turbulence modeling
The probability density function (PDF) transport equation method is a sophisticated model for the closure of turbulent mixing and turbulent reactive flows. An efficient solution approach for solving the PDF transport equation has been vital for the method to be widely used in applications. The Eulerian Monte Carlo fields (EMCF) method has been developed to solve the PDF transport equation efficiently for decades. A recent work by Wang et al. revealed a serious issue of the EMCF method for not being fully consistent with the PDF transport equation for which the method is designed to solve. This work advances the state of the art by introducing fully consistent EMCF methods for solving the PDF transport equation. The fully consistent EMCF formulations are derived for two different PDF equation forms. The consistency of the EMCF formulations is mathematically confirmed by examining the derived moment transport equations from the EMCF formulations and from the PDF transport equation. The method of manufactured solutions is employed to further verify the consistency and convergence of the different EMCF formulations numerically. The newly introduced EMCF formulations bring the EMCF method to full consistency with the PDF transport equations for the first time.
WE-AB-207A-02: John’s Equation Based Consistency Condition and Incomplete Projection Restoration Upon Circular Orbit CBCT
Purpose: In transmitted X-ray tomography imaging, projections are sometimes incomplete due to a variety of reasons, such as geometry inaccuracy, defective detector cells, etc. To address this issue, we have derived a direct consistency condition based on John’s Equation, and proposed a method to effectively restore incomplete projections based on this consistency condition. Methods: Through parameter substitutions, we have derived a direct consistency condition equation from John’s equation, in which the left side is only projection derivative of view and the right side is projection derivative of other geometrical parameters. Based on this consistency condition, a projection restoration method is proposed, which includes five steps: 1) Forward projecting reconstructed image and using linear interpolation to estimate the incomplete projections as the initial result; 2) Performing Fourier transform on the projections; 3) Restoring the incomplete frequency data using the consistency condition equation; 4) Performing inverse Fourier transform; 5) Repeat step 2)∼4) until our criteria is met to terminate the iteration. Results: A beam-blocking-based scatter correction case and a bad-pixel correction case were used to demonstrate the efficacy and robustness of our restoration method. The mean absolute error (MAE), signal noise ratio (SNR) and mean square error (MSE) were employed as our evaluation metrics of the reconstructed images. For the scatter correction case, the MAE is reduced from 63.3% to 71.7% with 4 iterations. Compared with the existing Patch’s method, the MAE of our method is further reduced by 8.72%. For the bad-pixel case, the SNR of the reconstructed image by our method is increased from 13.49% to 21.48%, with the MSE being decreased by 45.95%, compared with linear interpolation method. Conclusion: Our studies have demonstrated that our restoration method based on the new consistency condition could effectively restore the incomplete projections, especially for their high frequency component.
Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium
Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k{sub c}. For the limiting case of infinite correlation radius (k{sub c} = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k{sub c} value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k{sub c}, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k{sub c}/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k{sub c} values), because it describes both the shape and width of peaks much better than the standard approximation.
Eigenvalue computations with the QUAD4 consistent-mass matrix
The NASTRAN user has the option of using either a lumped-mass matrix or a consistent- (coupled-) mass matrix with the QUAD4 shell finite element. At the Sixteenth NASTRAN Users' Colloquium (1988), Melvyn Marcus and associates of the David Taylor Research Center summarized a study comparing the results of the QUAD4 element with results of other NASTRAN shell elements for a cylindrical-shell modal analysis. Results of this study, in which both the lumped-and consistent-mass matrix formulations were used, implied that the consistent-mass matrix yielded poor results. In an effort to further evaluate the consistent-mass matrix, a study was performed using both a cylindrical-shell geometry and a flat-plate geometry. Modal parameters were extracted for several modes for both geometries leading to some significant conclusions. First, there do not appear to be any fundamental errors associated with the consistent-mass matrix. However, its accuracy is quite different for the two different geometries studied. The consistent-mass matrix yields better results for the flat-plate geometry and the lumped-mass matrix seems to be the better choice for cylindrical-shell geometries.