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Results for “Fundamental nonlinearities in nature”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS2012

Robust Control Design for Uncertain Nonlinear Dynamic Systems

Robustness to parametric uncertainty is fundamental to successful control system design and as such it has been at the core of many design methods developed over the decades. Despite its prominence, most of the work on robust control design has focused on linear models and uncertainties that are non-probabilistic in nature. Recently, researchers have acknowledged this disparity and have been developing theory to address a broader class of uncertainties. This paper presents an experimental application of robust control design for a hybrid class of probabilistic and non-probabilistic parametric uncertainties. The experimental apparatus is based upon the classic inverted pendulum on a cart. The physical uncertainty is realized by a known additional lumped mass at an unknown location on the pendulum. This unknown location has the effect of substantially altering the nominal frequency and controllability of the nonlinear system, and in the limit has the capability to make the system neutrally stable and uncontrollable. Another uncertainty to be considered is a direct current motor parameter. The control design objective is to design a controller that satisfies stability, tracking error, control power, and transient behavior requirements for the largest range of parametric uncertainties. This paper presents an overview of the theory behind the robust control design methodology and the experimental results.

Kenny, Sean P.↗
DOE OSTI2020

Nonlinear propagating modes beyond the phonons in fluorite-structured crystals

The vibrational energy of crystals is known to propagate in a fixed number of phonon branches allowed by symmetry. In the realm of nonlinear dynamics, however, additional nonlinear propagating modes are possible. Nonlinear propagating modes have unique properties that are important in many disciplines including optical communications, conducting polymers, biology, magnetism, and nuclear physics. Yet, despite the crucial importance of crystal lattice vibrations in fundamental and applied science, such additional propagating modes have not been observed in ordinary crystals. Here, we show that propagating modes exist beyond the phonons in fluorite-structured thoria, urania, and natural calcium fluoride using neutron scattering and first-principles calculations. These modes are observed at temperatures ranging from 5 K up to 1200 K, extend to frequencies 30–40% higher than the maximum phonon frequency, and travel at velocities comparable to or higher than the fastest phonon. The nonlinear origin of the modes is explained in part via perturbation theory, which approximately accounts for nonlinearity. Given that these modes are still clearly observed at 5 K, they are likely an inherent feature of the quantum ground state. The existence of these waves in three-dimensional crystals may have ramifications for material properties.

36 MATERIALS SCIENCE↗
NASA NTRS1976

Optimization of multi-constrained structures based on optimality criteria

A weight-reduction algorithm is developed for the optimal design of structures subject to several multibehavioral inequality constraints. The structural weight is considered to depend linearly on the design variables. The algorithm incorporates a simple recursion formula derived from the Kuhn-Tucker necessary conditions for optimality, associated with a procedure to delete nonactive constraints based on the Gauss-Seidel iterative method for linear systems. A number of example problems is studied, including typical truss structures and simplified wings subject to static loads and with constraints imposed on stresses and displacements. For one of the latter structures, constraints on the fundamental natural frequency and flutter speed are also imposed. The results obtained show that the method is fast, efficient, and general when compared to other competing techniques. Extensions to the generality of the method to include equality constraints and nonlinear merit functions is discussed.

Rizzi, P.↗
NASA NTRS1972

Optimization of simple structures with higher mode frequency constraints.

Results of a study of least weight optimization of simple structures with a single natural frequency constraint. Variational techniques were used to derive the necessary equations, and numerical methods were used, where required, to find solutions to the resulting nonlinear, two-point boundary value problems. The numerical results suggest a similarity between the fixed fundamental frequency solution and solutions in which a single frequency other than fundamental is held fixed. Such a similarity is shown to occur under special circumstances, and the knowledge of the fundamental solution in these cases makes it possible to calculate the solution to problems with constraints on frequencies other than the fundamental.

Weisshaar, T. A.↗
NASA NTRS1997

Evolution of Structure in the Intergalactic Medium and the Nature of the LY-Alpha Forest

We have performed a detailed statistical study of the evolution of structure in a photoionized intergalactic medium (IGM) using analytical simulations to extend the calculation into the mildly nonlinear density regime found to prevail at z = 3. Our work is based on a simple fundamental conjecture: that the probability distribution function of the density of baryonic diffuse matter in the universe is described by a lognormal (LN) random field. The LN distribution has several attractive features and follows plausibly from the assumption of initial linear Gaussian density and velocity fluctuations at arbitrarily early times. Starting with a suitably normalized power spectrum of primordial fluctuations in a universe dominated by cold dark matter (CDM), we compute the behavior of the baryonic matter, which moves slowly toward minima in the dark matter potential on scales larger than the Jeans length. We have computed two models that succeed in matching observations. One is a nonstandard CDM model with OMEGA = 1, h = 0.5, and GAMMA = 0.3, and the other is a low-density flat model with a cosmological constant (LCDM), with OMEGA = 0.4, OMEGA(sub LAMBDA) = 0.6, and h = 0.65. In both models, the variance of the density distribution function grows with time, reaching unity at about z = 4, where the simulation yields spectra that closely resemble the Ly-alpha forest absorption seen in the spectra of high-z quasars. The calculations also successfully predict the observed properties of the Ly-alpha forest clouds and their evolution from z = 4 down to at least z = 2, assuming a constant intensity for the metagalactic UV background over this redshift range. However, in our model the forest is not due to discrete clouds, but rather to fluctuations in a continuous intergalactic medium. At z = 3; typical clouds with measured neutral hydrogen column densities N(sub H I) = 10(exp 13.3), 10(exp 13.5), and 10(exp 11.5) /sq cm correspond to fluctuations with mean total densities approximately 10, 1, and 0.1 times the universal mean baryon density. Perhaps surprisingly, fluctuations whose amplitudes are less than or equal to the mean density still appear as "clouds" because in our model more than 70% of the volume of the IGM at z = 3 is filled with gas at densities below the mean value.

Bi, Hongguang↗
NASA NTRS1986

Linear and weakly nonlinear aspects of free shear layer instability, roll-up, subharmonic interaction and wall influence

The growth of the momentum thickness and the modal disturbance energies are examined to study the nature and onset of nonlinearity in a temporally growing free shear layer. A shooting technique is used to find solutions to the linearized eigenvalue problem, and pseudospectral weakly nonlinear simulations of this flow are obtained for comparison. The roll-up of a fundamental disturbance follows linear theory predictions even with a 20 percent disturbance amplitude. A weak nonlinear interaction of the disturbance creates a finite-amplitude mean shear stress which dominates the growth of the layer momentum thickness, and the disturbance growth rate changes until the fundamental disturbance dominates. The fundamental then becomes an energy source for the harmonic, resulting in an increase in the growth rate of the subharmonic over the linear prediction even when the fundamental has no energy to give. Also considered are phase relations and the wall influence.

Cain, A. B.↗
NASA NTRS2019

Nonlinear Analysis of a Two-Parachute System Undergoing Pendulum Motion

Motion resembling that of a pendulum undergoing large-amplitude limit cycle oscillation was observed during a series of flight tests of an unoccupied Orion Capsule Parachute Assembly System (CPAS) comprised of two parachutes and a capsule payload. Large excursions away from vertical by the capsule could cause it to strike the ground or ocean at a large angle with respect to vertical, or at a large horizontal speed. These conditions are undesirable because they would endanger the occupants of the capsule in an actual mission. A simplified planar dynamics model in conjunction with a nonlinear normal force coefficient vs. angle of attack model serves as the basis of an analytical investigation of the fundamental dynamics of this pendulum motion. Output error methodology from system identification theory was used to identify the parameters of the nonlinear aerodynamics model. The identified model yielded excellent comparison with portions of flight test data where the pendulum motion occurred. Due to the inherent nonlinear nature of the pendulum motion limit cycle, traditional nonlinear analysis techniques were applied to gain further insight into the system. Lyapunov’s direct method provided mathematical proof in the absolute stability of the pendulum mode. Describing Function method was used to predict the amplitude and frequency of the limit cycle oscillation. Finally, phase plane analysis allowed easy visualization on the size and shape of the limit cycle with respect to variations in key aerodynamic parameters.

Pei, Jing↗
DOE OSTI2026

Phenomena-based graph representations and applications to chemical process simulation

Rapid and robust simulation of chemical processes is critical to conduct process design, optimization, techno-economic analysis, and sustainability analysis. Yet, efficiently solving simulation models remains a challenge due to the highly coupled and nonlinear nature of the underlying algebraic equations that capture the physical phenomena taking place in the process (e.g., material and energy conservation, phase equilibrium, reactions). In this work, we show that graph-theoretic representations of the physical phenomena within unit operations can help navigate and decompose equations to systematically identify alternative approaches for fast and robust numerical solutions. Specifically, we present a graph-theoretic abstraction that captures the connectivity between the model variables/equations and use this abstraction to group variables/equations into fundamental phenomena. We show that phenomena-based decomposition of the underlying equations can help decouple nonlinearities and enforce material/energy conservation at the process level to accelerate convergence. The proposed decomposition approach differs from the more traditional sequential modular simulation approach, in which equations are grouped and decomposed by unit operations. We implemented the phenomena-based decomposition in BioSTEAM—an open-source process simulation platform in Python—and demonstrated that this approach can converge a variety of separation process models. Compared to sequential modular simulation, the phenomena-based approach can converge idealized systems faster, but it can be slower for (or even fail to converge) highly coupled and nonideal process systems.

Convergence↗
NASA NTRS2017

Unravelling the Components of a Multi-Thermal Coronal Loop Using Magnetohydrodynamic Seismology

Coronal loops, constituting the basic building blocks of the active Sun, serve as primary targets to help understand the mechanisms responsible for maintaining multi-million Kelvin temperatures in the solar and stellar coronae. Despite significant advances in observations and theory, our knowledge on the fundamental properties of these structures is limited. Here, we present unprecedented observations of accelerating slow magnetoacoustic waves along a coronal loop that show differential propagation speeds in two distinct temperature channels, revealing the multi-stranded and multithermal nature of the loop. Utilizing the observed speeds and employing nonlinear force free magnetic field extrapolations, we derive the actual temperature variation along the loop in both channels, and thus are able to resolve two individual components of the multithermal loop for the first time. The obtained positive temperature gradients indicate uniform heating along the loop, rather than isolated foot point heating.

Prasad, S. Krishna↗
DOE OSTI2022

The principle of minimal power

This article is devoted to the memory of Yuri P Raizer, who passed away in 2021. He left a noticeable trace in gas discharge physics. The principle of minimal power (the state that requires minimal power is most probable) is thoroughly used in his books. Although the fundamental laws of physics do not imply this ad hoc principle, a detailed analysis of underlying phenomena can often reveal why nature prefers this path. Raizer illustrated this principle for plasma stratification, formation of electrode spots, discharge constriction, the shape of an arc channel, etc. We argue that the nonlinearity of equations describing gas discharges can often justify the realization of a plasma state maintained at minimal electric power. This nonlinearity appears because small groups of energetic electrons often control the ionization processes. The number of these electrons depends strongly on the ratio of the electric field to gas density, E/N. Under certain conditions, the ionization rate can also depend nonlinearly on electron density due to stepwise ionization and Coulomb collisions. Here, we use the principle of minimal power to illustrate some of Raizer's contributions to gas discharge physics from a single point of view. We demonstrate that nonlinearity of ionization processes in gas discharges can substantiate this principle for plasma stratification. However, striations of s, p, and r types in neon could exist with minimal or no ionization enhancement. This reminds us of Raizer's warning that applying the minimal power principle could lead to erroneous predictions, and a proper theory is required in each case to justify its use. 'The phenomenon of striations satisfies the principle of minimal power' – Yuri Raizer

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2003

The Effect of Surface Topography on the Nonlinear Dynamics of Rossby Waves

Boussinesq convection in rotating systems attracts a sustained attention of the fluid dynamics community, because it has intricate non-linear dynamics (Cross & Hohenberg 1993) and plays an important role in geophysical and astrophysical applications, such as the motion of the liquid outer core of Earth, the Red Spot in Jupiter, the giant cells in the Sun etc. (Alridge et al. 1990). A fundamental distinction between the real geo- and astrophysical problems and the idealized laboratory studies is that natural systems are inhomogeneous (Alridge et al. 1990). Heterogeneities modulate the flow and influence significantly the dynamics of convective patterns (Alridge et al. 1990; Hide 1971). The effect of modulations on pattern formation and transition to turbulence in Boussinesq convection is far from being completely understood (Cross & Hohenberg 1993; Aranson & Kramer 2002). It is generally accepted that in the liquid outer core of the Earth the transport of the angular momentum and internal heat occurs via thermal Rossby waves (Zhang et al. 2001; Kuang & Bloxham 1999). These waves been visualized in laboratory experiments in rotating liquid-filled spheres and concentric spherical shells (Zhang et al. 2001; Kuang & Bloxham 1999). The basic dynamical features of Rossby waves have been reproduced in a cylindrical annulus, a system much simpler than the spherical ones (Busse & Or 1986; Or & Busse 1987). For convection in a cylindrical annulus, the fluid motion is two-dimensional, and gravity is replaced by a centrifugal force, (Busse & Or 1986; Or & Busse 1987). Hide (1971) has suggested that the momentum and heat transport in the core might be influenced significantly by so-called bumps, which are heterogeneities on the mantle-core boundary. To model the effect of surface topography on the transport of momentum and energy in the liquid outer core of the Earth, Bell & Soward (1996), Herrmann & Busse (1998) and Westerburg & Busse (2001) have studied the nonlinear dynamics of thermal Rossby waves in a cylindrical annulus with azimuthally modulated height.

Abarzhi, S. I.↗
DOE OSTI2023

Data-driven prediction of α IIb β 3 integrin activation paths using manifold learning and deep generative modeling

The integrin heterodimer is a transmembrane protein critical for driving cellular process and is a therapeutic target in the treatment of multiple diseases linked to its malfunction. Activation of integrin involves conformational transitions between bent and extended states. Some of the conformations that are intermediate between bent and extended states of the heterodimer have been experimentally characterized, but the full activation pathways remain unresolved both experimentally due to their transient nature and computationally due to the challenges in simulating rare barrier crossing events in these large molecular systems. An understanding of the activation pathways can provide new fundamental understanding of the biophysical processes associated with the dynamic interconversions between bent and extended states and unveil new putative therapeutic targets. In this work, we apply nonlinear manifold learning to coarse-grained molecular dynamics simulations of bent, extended, and two intermediate states of αI I b β3 integrin to learn a low-dimensional embedding of the configurational phase space. We then train deep generative models to learn an inverse mapping between the low-dimensional embedding and high-dimensional molecular space and use these models to interpolate the molecular configurations constituting the activation pathways between the experimentally characterized states. Furthermore, this work furnishes plausible predictions of integrin activation pathways and reports a generic and transferable multi-scale technique to predict transition pathways for biomolecular systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Dark Matter Constraints from Small-Scale Cosmic Structure

Small-scale cosmic structure provides a powerful test of the fundamental nature of dark matter (DM). A wide range of DM models impact matter clustering on small scales, including warm, fuzzy, and (self-)interacting DM. In these scenarios, DM physics such as free-streaming, wave interference, and self/Standard Model interactions alter the abundance and internal structure of DM halos. Cosmological and astrophysical probes of nonlinear structure---including dwarf galaxies, strong lensing, the Lyman-$α$ forest, stellar streams, and high-redshift galaxies---are therefore sensitive to these effects. Here, we review DM constraints provided by small-scale structure, focusing on observables that probe scales smaller than $\sim 1~\mathrm{Mpc}$, which define the frontier of current measurements. We summarize how these constraints have been translated to limits on microphysical DM models, and we discuss key modeling uncertainties and observational systematics. Finally, we highlight the growing importance of probe combination and simulation-based inference for this field, and we overview upcoming observational facilities that will sharpen small-scale structure tests of DM physics.

Nadler, Ethan O. [UC, San Diego] (ORCID:0000000211↗
NASA NTRS1995

Static stability of a three-dimensional space truss

In order to deploy large flexible space structures it is necessary to develop support systems that are strong and lightweight. The most recent example of this aerospace design need is vividly evident in the space station solar array assembly. In order to accommodate both weight limitations and strength performance criteria, ABLE Engineering has developed the Folding Articulating Square Truss (FASTMast) support structure. The FASTMast is a space truss/mechanism hybrid that can provide system support while adhering to stringent packaging demands. However, due to its slender nature and anticipated loading, stability characterization is a critical part of the design process. Furthermore, the dire consequences surely to result from a catastrophic instability quickly provide the motivation for careful examination of this problem. The fundamental components of the space station solar array system are the (1) solar array blanket system, (2) FASTMast support structure, and (3) mast canister assembly. The FASTMast once fully deployed from the canister will provide support to the solar array blankets. A unique feature of this structure is that the system responds linearly within a certain range of operating loads and nonlinearly when that range is exceeded. The source of nonlinear behavior in this case is due to a changing stiffness state resulting from an inability of diagonal members to resist applied loads. The principal objective of this study was to establish the failure modes involving instability of the FASTMast structure. Also of great interest during this effort was to establish a reliable analytical approach capable of effectively predicting critical values at which the mast becomes unstable. Due to the dual nature of structural response inherent to this problem, both linear and nonlinear analyses are required to characterize the mast in terms of stability. The approach employed herein is one that can be considered systematic in nature. The analysis begins with one and two-dimensional failure models of the system and its important components. From knowledge gained through preliminary analyses a foundation is developed for three-dimensional analyses of the FASTMast structure. The three-dimensional finite element (FE) analysis presented here involves a FASTMast system one-tenth the size of the actual flight unit. Although this study does not yield failure analysis results that apply directly to the flight article, it does establish a method by which the full-scale mast can be evaluated.

Shaker, John F.↗
DOE OSTI2021

Rayleigh–Taylor and Richtmyer–Meshkov instabilities: A journey through scales

Hydrodynamic instabilities such as Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities usually appear in conjunction with the Kelvin–Helmholtz (KH) instability and are found in many natural phenomena and engineering applications. They frequently result in turbulent mixing, which has a major impact on the overall flow development and other effective material properties. This can either be a desired outcome, an unwelcome side effect, or just an unavoidable consequence, but must in all cases be characterized in any model. The RT instability occurs at an interface between different fluids, when the light fluid is accelerated into the heavy. The RM instability may be considered a special case of the RT instability, when the acceleration provided is impulsive in nature such as that resulting from a shock wave. Here in this pedagogical review, we provide an extensive survey of the applications and examples where such instabilities play a central role. First, fundamental aspects of the instabilities are reviewed including the underlying flow physics at different stages of development, followed by an overview of analytical models describing the linear, nonlinear and fully turbulent stages. RT and RM instabilities pose special challenges to numerical modeling, due to the requirement that the sharp interface separating the fluids be captured with fidelity. These challenges are discussed at length here, followed by a summary of the significant progress in recent years in addressing them. Examples of the pivotal roles played by the instabilities in applications are given in the context of solar prominences, ionospheric flows in space, supernovae, inertial fusion and pulsed-power experiments, pulsed detonation engines and Scramjets. Progress in our understanding of special cases of RT/RM instabilities is reviewed, including the effects of material strength, chemical reactions, magnetic fields, as well as the roles the instabilities play in ejecta formation and transport, and explosively expanding flows. The article is addressed to a broad audience, but with particular attention to graduate students and researchers who are interested in the state-of-the-art in our understanding of the instabilities and the unique issues they present in the applications in which they are prominent.

42 ENGINEERING↗
NASA NTRS1978

A three-dimensional model of corotating streams in the solar wind. I - Theoretical foundations

The paper is concerned with the development of the theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 Rs where the flow is supersonic) is approximately described by the nonlinear, single-fluid, polytropic magnetohydrodynamic or hydrodynamic equations. Efficient numerical techniques are outlined for solving this complex system of coupled, hyperbolic partial differential equations. The present formulation is inviscid and nonmagnetic, but the methods used allow for the potential inclusion of both features with only modest modifications. A simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution. It is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction. Comparison with simpler models demonstrates the essential nonlinear, multidimensional nature of the interplanetary dynamics.

Pizzo, V.↗
DOE OSTI2024

VpROM: a novel variational autoencoder-boosted reduced order model for the treatment of parametric dependencies in nonlinear systems

Reduced Order Models (ROMs) are of considerable importance in many areas of engineering in which computational time presents difficulties. Established approaches employ projection-based reduction, such as Proper Orthogonal Decomposition. The limitation of the linear nature of such operators is typically tackled via a library of local reduction subspaces, which requires the assembly of numerous local ROMs to address parametric dependencies. Our work attempts to define a more generalisable mapping between parametric inputs and reduced bases for the purpose of generative modeling. We propose the use of Variational Autoencoders (VAEs) in place of the typically utilised clustering or interpolation operations, for inferring the fundamental vectors, termed as modes, which approximate the manifold of the model response for any and each parametric input state. The derived ROM still relies on projection bases, built on the basis of full-order model simulations, thus retaining the imprinted physical connotation. However, it additionally exploits a matrix of coefficients that relates each local sample response and dynamics to the global phenomena across the parametric input domain. The VAE scheme is utilised for approximating these coefficients for any input state. This coupling leads to a high-precision low-order representation, which is particularly suited for problems where model dependencies or excitation traits cause the dynamic behavior to span multiple response regimes. Moreover, the probabilistic treatment of the VAE representation allows for uncertainty quantification on the reduction bases, which may then be propagated to the ROM response. The performance of the proposed approach is validated on an open-source simulation benchmark featuring hysteresis and multi-parametric dependencies, and on a large-scale wind turbine tower characterised by nonlinear material behavior and model uncertainty.

Conditional VAEs↗
NASA NTRS1999

Bidirectional Reflectance of a Macroscopically Flat, High-Albedo Particulate Surface: An Efficient Radiative Transfer Solution and Applications to Regoliths

Many remote sensing applications rely on accurate knowledge of the bidirectional reflection function (BRF) of surfaces composed of discrete, randomly positioned scattering particles. Theoretical computations of BRFs for plane-parallel particulate layers are usually reduced to solving the radiative transfer equation (RTE) using one of existing exact or approximate techniques. Since semi-empirical approximate approaches are notorious for their low accuracy, violation of the energy conservation law, and ability to produce unphysical results, the use of numerically exact solutions of RTE has gained justified popularity. For example, the computation of BRFs for macroscopically flat particulate surfaces in many geophysical publications is based on the adding-doubling (AD) and discrete ordinate (DO) methods. A further saving of computer resources can be achieved by using a more efficient technique to solve the plane-parallel RTE than the AD and DO methods. Since many natural particulate surfaces can be well represented by the model of an optically semi-infinite, homogeneous scattering layer, one can find the BRF directly by solving the Ambartsumian's nonlinear integral equation using a simple iterative technique. In this way, the computation of the internal radiation field is avoided and the computer code becomes highly efficient and very accurate and compact. Furthermore, the BRF thus obtained fully obeys the fundamental physical laws of energy conservation and reciprocity. In this paper, we discuss numerical aspects and the computer implementation of this technique, examine the applicability of the Henyey-Greenstein phase function and the sigma-Eddington approximation in BRF and flux calculations, and describe sample applications demonstrating the potential effect of particle shape on the bidirectional reflectance of flat regolith surfaces. Although the effects of packing density and coherent backscattering are currently neglected, they can also be incorporated. The FORTRAN implementation of the technique is available on the World Wide Web, and can be applied to a wide range of remote sensing problems. BRF computations for undulated (macroscopically rough) surfaces are more complicated and often rely on time consuming Monte Carlo procedures. This approach is especially inefficient for optically thick, weakly absorbing media (e.g., snow and desert surfaces at visible wavelengths since a photon may undergo many internal scattering events before it exists the medium or is absorbed. However, undulated surfaces can often be represented as collections of locally flat tilted facets characterized by the BRF found from the traditional plane parallel RTE. In this way the MOnte Carlo procedure could be used only to evaluate the effects of surface shadowing and multiple surface reflections, thereby bypassing the time-consuming ray tracing inside the medium and providing a great savings of CPU time.

Mishchenko, Michael I.↗