Logarithmic Interpolation Research Articles

A new method for CBR prediction using fuzzy set theory

California Bearing Ratio (CBR) values of the subgrade soils are key parameters in highway route survey and project design. However, in order to calculate the CBR value, time-consuming and specialized tests must be performed in the laboratory or in the field. In this study, a model based on Fuzzy set theory is proposed to predict the CBR value of soils. The geological characteristics of the fine-sized aggregate in soils are considered as the key parameter in the prediction model.For this purpose, novel Fuzzy membership functions are defined. Logarithmic interpolation approach, which takes into account the Atterberg limits and Proctor parameters of the soils, was used to determine the membership degrees of the gravel and sand sized aggregates forming the soil. The membership degrees of fine-sized aggregates were determined by basic sigmoid membership function according to the type of clay they contain. In order to verify the accuracy of the model, the calculated results were compared with the soaked CBR values obtained from laboratory tests and it was found that there was a high correlation (R2=0.86) between each other. Considering the obtained results together, it was determined that the effect of variables such as the geological class to which the aggregates forming the soil belong, which have the potential to affect the CBR value of the soils, on the results of the CBR prediction models can be eliminated by means of membership functions.

Near‐surface wind profiles from numerical model predictions. Part I: Algorithms and comparisons with wind profile based on Monin–Obukhov similarity theory

AbstractWinds are predicted on the discrete grid of numerical weather and climate models. Winds distribute nonlinearly on the height in the near‐surface layer, and a 10 m wind prediction within the layer is often diagnosed upon the Monin–Obukhov similarity theory flux–profile relationship determined from winds at the lowest grid level, the near‐surface atmospheric stability, and surface properties, which leads to concerns that systemic biases may be introduced to the diagnosed wind. Algorithms are proposed to derive near‐surface wind profiles from the grid‐based numerical model forecasts at multiple model levels under the framework of momentum conservations with an implicit solution, associated with simple logarithmic plus linear interpolation in exceptional exemptional conditions. The diagnosed wind profile coheres to the model prediction at the grid level and exhibits differences from the profile using the conventional scheme in the quasi‐steady thermal stratification and non‐steady transitional conditions, retreating to the same logarithmic profile in the neutral condition.

Corrigendum to “Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results”

This corrigendum corrects several mistakes in Appendix B.3 of [Ann. Inst. H. Poincaré Anal. Non Linéaire (2023). .

Global existence and exponential decay of the 2D density‐dependent Bénard system with vacuum in bounded domain

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained.

Pull-off of viscoelastic spherical contact

This paper studied the adhesive contact between a viscoelastic sphere and a rigid plane, and presented a dimensionless first-order differential equation to directly describe the relationship between the applied load and the contact radius. By adopting a combination strategy, the singularity of Muller’s dimensionless first-order differential equation was avoided well. To study the dependence of the pull-off force on the initial applied load, a table-based linear and logarithmic interpolation method was presented to determine the contact radius at the pull-off. The dependence of the square root (ζmax) of the ratio of the effective adhesion work at the upper bound to the equilibrium adhesion work on Muller’s parameter β and material constant n was deduced theoretically. And the relationship between the contact radius at the upper bound and ζmax was also deduced theoretically. In addition, the ratio of the pull-off force to the maximum pull-off force was determined to be cubic function of the ratio of the contact radius at the pull-off to that at the upper bound numerically. If β, n and the initial applied load were given, utilizing these relationships, the pull-off force and the maximum pull-off force could be predicted well. Lastly, Violano-Chateauminois-Afferrante JKR-like formula for the pull-off force and the upper bound was verified.

Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results

We consider Gagliardo–Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.

Logarithm-Based Methods for Interpolating Quaternion Time Series

In this paper, we discuss a modified quaternion interpolation method based on interpolations performed on the logarithmic form. This builds on prior work that demonstrated this approach maintains C2 continuity for prescriptive rotation. However, we develop and extend this method to descriptive interpolation, i.e., interpolating an arbitrary quaternion time series. To accomplish this, we provide a robust method of taking the logarithm of a quaternion time series such that the variables θ and n^ have a consistent and continuous axis-angle representation. We then demonstrate how logarithmic quaternion interpolation out-performs Renormalized Quaternion Bezier interpolation by orders of magnitude.

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and Lp−Lq estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results.

Global well-posedness to the Cauchy problem of 2D nonhomogeneous Bénard system with large initial data and vacuum

This paper establishes the global well-posedness of strong solutions to the nonhomogeneous Bénard system with positive density at infinity in the whole space R2. We obtain the global existence and uniqueness of strong solutions for general large initial data. Our method relies on the dedicate energy estimates and a logarithmic interpolation inequality.

An elastoplastic model for anisotropic clays with improved yield surfaces and rotational hardening rule

Soil is a multiphase material with irregularly shaped particles, the behaviour of which is highly nonlinear, anisotropic, and inelastic. Anisotropy and its evolution characteristics are crucial in soil’s constitutive modelling, particularly for clayey soil. A new anisotropic constitutive model for remoulded clay is proposed based on improved yield surfaces and rotational hardening rules for initial and induced anisotropy. The logarithmic interpolation function is adopted to describe the volumetric deformation characteristics of clay in an arbitrary stress state, and a flexible yield surface shape is obtained by changing the newly introduced parameter n. This new parameter can be determined by fitting the undrained shear strength or stiffness of stress–strain curves. A key feature of the proposed evolutional rule for anisotropy is that the principal direction of the yield surfaces in the stress plane can be consistent with the isotropic stress axis when the critical state is reached. The proposed elastoplastic model shows good capacity for describing laboratory results from oedometer tests with constant stress ratios, reconsolidation tests with various stress ratios, triaxial compression tests, and triaxial extension tests under both undrained and drained conditions.

Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

In dimension two, we investigate a free energy and the ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrodinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in$\mathbb {R}^{d}$with$d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in$\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in$\mathbb {R}^{3}$by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

We establish global well-posedness of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with non-negative density on the whole space $${\mathbb {R}}^2$$ . More precisely, under compatibility conditions for the initial data, we show the global existence and uniqueness of strong solutions. Our method relies on delicate energy estimates and a logarithmic interpolation inequality. In particular, the initial data can be arbitrarily large.

Quantitative relationship of fundamental rheological properties of bitumen with the empirical Ring and Ball softening point

This paper employs the linear viscoelastic parameter G′/tanδ as the quantitative criterion for identifying a common relationship with the Ring and Ball softening point of both unmodified and polymer-modified bitumen (PMB). Temperature sweep was conducted at 10 rad/s with dynamic shear rheometer to determine the complex shear modulus G* and phase angle δ of bitumen. The parameter G′/tanδ could be obtained. Logarithmic interpolation was applied for G′/tanδ to determine the equivalent softening temperature Ts _G′/tanδ at which G′/tanδ equals the proposed criterion values. With this method, a good agreement could be reached between Ts _G′/tanδ and the softening point for both unmodified bitumen and PMB. While keeping high accuracy for unmodified bitumen, the accuracy of Ts _G′/tanδ as an estimation of the softening point of PMB was largely improved when compared to the iso-modulus approach. The reliability of this quantitative relationship is preliminarily verified and its implementation is discussed in the Black space.

Перспективы увеличения минеральной базы цветной металлургии

The importance of research into enhancing the metal mining technologies is justified by the declining availability of mineral resources due to non-compliance of the conventional mining methods with the market conditions. The traditional ore processing technologies are accompanied by accumulation of processing tailings. Zero-waste recycling of primary processing waste, which is often used without extracting metals to match the sanitary standards, is not evolving. The acute environmental issues are primarily exacerbated by a lack of levers for centralized accounting and management of the accumulated waste. The aim of the study is to develop new technologies that are optimized in terms of complete utilization of the off-grade raw materials i.e. wastes from primary ore processing. The effectiveness of leaching technologies is proved with a complex method that involves experiments and calculations comparing the performance of processing options using the Box-Behnken design of metal extraction and interpreting the results in the form of logarithmic or polynomial interpolation. Quantitative values were obtained and cross-plots of metal extraction dependence on the contributing factors were made, which allow characterizing the leaching processes of polymetals and ferruginous quartzites in the disintegrator. It has been proved that mechanochemical treatment provides higher metal yield (up to 45%) than traditional waste processing technologies while securing a safety level that meets the sanitary requirements. It has also been determined that disintegrator activation during the metal leaching process increases the strength of concrete mixtures based on re-treated tailings, both as aggregates and as a binder. A conclusion is made that activation of the leaching processes in a disintegrator ensures the extraction of 50 to 80% of metals that are not available for extraction from mill tailing with conventional technologies. Development of man-made deposits using innovative technologies based on metal leaching is a real step towards expanding the mineral resource base of the metallurgical industry and improving the environmental situation in the mining regions. In contrast to technologies of a similar scope and purpose, the proposed technology makes it possible to process ores in a zero-waste manner without creating new tailings.

An evolutionary, data-driven approach for mechanism optimization: theory and application to ammonia combustion

In this work, we propose a novel data-driven approach for detailed kinetic mechanisms optimization. The approach is founded on a curve matching-based objective function and includes a methodology for the optimisation of pressure-dependent reactions via logarithmic interpolation (PLOG format). In order to highlight the advantages of the new formulation of the objective function, a comparison with L1 and L2 norm is performed. The selection of impactful reactions is carried out by introducing a Cumulative Impact Function (CIF), while an Evolutionary Algorithm (EA) is adopted for the optimization. The capabilities of the proposed methodology were demonstrated using a database of ~635 experimental datapoints on ammonia combustion, covering standard targets like ignition delay times, speciation and laminar flame speed. The optimization was carried out starting from a recently published mechanism, describing ammonia pyrolysis and oxidation, largely developed using first-principles calculation of rate constants. After the selection of the 24 most impactful reactions, the related 101 normalized Arrhenius parameters were simultaneously varied, within their uncertainty bounds. Their uncertainty bounds were taken from the literature, when available, or estimated according to the level of theory adopted for the determination of the rate constant. Hence, we also provide guidelines to estimate uncertainty for reaction rate constants derived from first principles calculations using well consolidated computational protocols as a reference. The optimized mechanism was found to improve the nominal one, showing a satisfactory agreement over the entire range of operating conditions. Moreover, the use of ‘curve matching’ indices was found to outperform the adoption of L1 and L2 norms. The comparison between the nominal mechanism and the one optimized via curve matching allowed a clear identification of different critical reaction pathways for different experimental targets. From this perspective, the methodology proposed herein can find further application as a useful design-of-experiments tool for an accurate evaluation of crucial kinetic constants, thus driving further mechanism improvement.

Optimality of Serrin type extension criteria to the Navier-Stokes equations

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ L θ (0, T; U˙ −α ∞,1/θ,∞) for 2/θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T; V˙ 0 ∞,∞,2 ) , where U˙ s p,β,σ and V˙ s p,q,θ are Banach spaces that may be larger than the homogeneous Besov space B˙ s p,q. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

Phase field-based topology optimization of metallic structures for microwave applications using adaptive mesh refinement

An important issue in the metallic structure design for microwave applications is the skin depth which may significantly degrade the intended performance. In order to address the skin depth issue, a perfect electric conductor or an impedance boundary condition can be imposed on the outer surface of the metallic structure. However, it is hard to impose such boundary conditions in topology optimization-based design because clear boundary definition with the best performance is difficult in ordinary topology optimization. To overcome such difficulties, this study proposes the phase field-based topological design process of the metallic structure using the adaptive mesh refinement for improved boundary representation. The reaction-diffusion equations combined with the double-well potentials are used as the update scheme. An exponential function using logarithmic interpolation with penalization is used for electrical conductivity definition of gray materials to replace the PEC boundary conditions for the metallic structure with the adaptive mesh refinement. The design process consists of two steps; the first step derives a roughly optimized shape which is used for the initial shape in the next step, where the finally optimized shape is derived using adapted mesh with respect to the material distribution. The validity of the proposed approach is confirmed through three numerical examples.

The Equivalence Theorem for Logarithmic Interpolation Spaces in the Quasi-Banach Case

We study the description by means of the J -functional of logarithmic interpolation spaces (A_0, A_1)_{1,q,\mathbb{A}} in the category of the p -normed quasi-Banach couples ( 0 &lt; p \leq 1 ). When (A_0, A_1) is a Banach couple, it is known that the description changes depending on the relationship between q and \mathbb{A} . In our more general setting, the parameter p also has an important role as the results show.

Incorporating a Canopy Parameterization within a Coupled Fire-Atmosphere Model to Improve a Smoke Simulation for a Prescribed Burn

Forecasting fire growth, plume rise and smoke impacts on air quality remains a challenging task. Wildland fires dynamically interact with the atmosphere, which can impact fire behavior, plume rises, and smoke dispersion. For understory fires, the fire propagation is driven by winds attenuated by the forest canopy. However, most numerical weather prediction models providing meteorological forcing for fire models are unable to resolve canopy winds. In this study, an improved canopy model parameterization was implemented within a coupled fire-atmosphere model (WRF-SFIRE) to simulate a prescribed burn within a forested plot. Simulations with and without a canopy wind model were generated to determine the sensitivity of fire growth, plume rise, and smoke dispersion to canopy effects on near-surface wind flow. Results presented here found strong linkages between the simulated fire rate of spread, heat release and smoke plume evolution. The standard WRF-SFIRE configuration, which uses a logarithmic interpolation to estimate sub-canopy winds, overestimated wind speeds (by a factor 2), fire growth rates and plume rise heights. WRF-SFIRE simulations that implemented a canopy model based on a non-dimensional wind profile, saw significant improvements in sub-canopy winds, fire growth rates and smoke dispersion when evaluated with observations.