Nonlinear Shape Research Articles

Post-Miocene topographic rejuvenation in an elevated passive continental margin not characterized by a sharp escarpment (northern end of the Mantiqueira Range, Brazil)

A sharp escarpment lying close to the coast or more in the continental interior defines the morphology of Elevated Passive Continental Margins (EPCMs). Most studies on the dynamics and evolution of EPCMs, including numerical modeling exercises, concentrate on the geometry and evolution of escarpments located at or near a continental drainage divide. However, topographic relief varies considerably along the length of an EPCM, as one can observe in southeastern Brazil and southeastern Australia, and in some cases, the steep, wall-like escarpment may not be present. Here we investigate the post-rift geomorphic evolution in an EPCM lacking a sharp escarpment (southeastern Brazil), focusing on exploring topographic rejuvenation and links with potential controls such as the arrangement of pre-existing structures and lithological variability. We present topographic data showing a regional zone of high channel steepness and local relief extending continuously through the middle part of all seaward-dipping catchments. In this regional “belt” of high topography lies many knickpoints, and all rivers crossing it are marked by non-linear shapes in elevation-χ space. The structural data show a large number of NE-SW and NW-SE oriented brittle structures with slickensides recorded on filling materials following the orientation of pre-existing basement structures, suggesting a multiphase brittle deformation with at least two main paleostress tectonic regimes between the Miocene and Pleistocene. Our results indicate post-Miocene topographic rejuvenation and brittle deformation, probably driven by the accumulation of far-field stresses in pre-existing weakness zones. Nevertheless, the origin of the regional drainage divide and adjacent areas, located considerably upstream of the regional knickzone, must predate the topographic resurgence we infer, implying that older and more recent topographic elements coexist in the landscape. Our study demonstrates that post-rift topographic rejuvenation (and brittle deformation) in an EPCM is not necessarily linked to the formation or persistence of a “Great Escarpment”.

Press-braked stainless steel channel sections under major-axis combined loading: Tests, simulations and design

This paper reports an experimental and numerical investigation into the cross-section behaviour and resistances of press-braked stainless steel channel sections under combined compression and major-axis bending moment. A testing programme was firstly conducted, including initial local geometric imperfection measurements and major-axis eccentric compression tests on ten press-braked stainless steel channel section stub column specimens. Following the testing programme, a numerical modelling programme was performed, where finite element models were developed and validated against the test results and then adopted to perform parametric studies to generate further numerical data over a wide range of cross-section dimensions and initial loading eccentricities. Based on the obtained test and numerical data, the existing design interaction curves, as given in the European code, American specification and Australian/New Zealand standard, were assessed and shown to result in conservative cross-section resistance predictions for press-braked stainless steel channel sections under major-axis combined loading. The conservatism of the codified design interaction curves stems from the conservative end points (i.e. cross-section resistances under pure compression and pure bending), which are determined without considering the effect of material strain hardening, and the inefficient linear shape, which ignores the effect of stress redistribution. Improved design interaction curves were proposed through the adoption of (i) more accurate end points, which are calculated with a rational exploitation of material strain hardening by continuous strength method and (ii) more efficient nonlinear shape, which allows for the stress redistribution. The proposed design interaction curves were shown to result in substantially more accurate resistance predictions for press-braked stainless steel channel sections under major-axis combined loading than their codified counterparts. The reliability of the proposed design interaction curves was confirmed by means of statistical analyses.

What Do Prosumer Marginal Utility Functions Look Like? Derivation and Analysis

Marginal Utility Functions (MUFs) encapsulate the prosumer willingness to trade energy with other market agents. In large-scale distributed optimization schemes scalability and convergence are crucial, and the assumption of MUFs is common. In this work, instead of assuming the shape and coefficients of those functions, a method is proposed to derive them based on the optimization of a prosumer's energy procurement problem. We formulate a rolling-horizon optimization problem that considers asset characteristics and network tariffs, and we utilize forecasts to capture the effects of uncertainty. We use this formulation to calculate the true prosumer MUF. A test case with real data is used to investigate the shape of these functions. Our results reveal that they present a non-linear shape under certain conditions, they cannot be a priori derived, and their form is sensitive to a variety of factors. This indicates that MUFs should be constructed based on an approach similar to the one proposed in this paper to more accurately capture the prosumer's true willingness to trade. A linearized version of these functions, which is computationally attractive and scalable, can be determined using a least-squares estimator that achieves the best fit to the prosumer's preferences under a linearity constraint.

Non-linear shape oscillations of a viscous liquid drop

Non-linear shape oscillations of a viscous liquid drop

Nonlinear Oscillations and Stability of Rotating Low-Aspect-Ratio Blades Containing Internal Resonance

Rotating orthotropic blade equations based on the assumptions of the plate theory and the nonlinear von Kármán strain–displacement relations are obtained, and, furthermore, updated equations associated with prestressed configuration are provided. The extended Galerkin method is employed to determine the prestressed configuration due to the rotation and to obtain the linear frequencies and normal modes of updated equations. The multiple scales method, in combination with the Galerkin method, is used to study the nonlinear vibrations and to obtain analytical relations for the nonlinear frequencies and mode shapes. The role of different parameters on the backbone curves and effective nonlinearity coefficients is evaluated. In the nonlinear analysis, internal resonance is also considered. Moreover, the fourth-order Runge–Kutta method is applied to numerically analyze the nonlinear responses. A detailed investigation shows that in the torsion modes, the aspect ratio and the stiffness ratio changes influence the nonlinearity type and quality of the results. The quantities of effective nonlinearity coefficients depend on the rotational speed and the types of material and geometry. In the case of internal resonance, the stability of the nonlinear modes has been evaluated, and it is found that the rotating blades have different stable and unstable coupled modes.

Machine learning identifies ICU outcome predictors in a multicenter COVID-19 cohort

BackgroundIntensive Care Resources are heavily utilized during the COVID-19 pandemic. However, risk stratification and prediction of SARS-CoV-2 patient clinical outcomes upon ICU admission remain inadequate. This study aimed to develop a machine learning model, based on retrospective & prospective clinical data, to stratify patient risk and predict ICU survival and outcomes.MethodsA Germany-wide electronic registry was established to pseudonymously collect admission, therapeutic and discharge information of SARS-CoV-2 ICU patients retrospectively and prospectively. Machine learning approaches were evaluated for the accuracy and interpretability of predictions. The Explainable Boosting Machine approach was selected as the most suitable method. Individual, non-linear shape functions for predictive parameters and parameter interactions are reported.Results1039 patients were included in the Explainable Boosting Machine model, 596 patients retrospectively collected, and 443 patients prospectively collected. The model for prediction of general ICU outcome was shown to be more reliable to predict “survival”. Age, inflammatory and thrombotic activity, and severity of ARDS at ICU admission were shown to be predictive of ICU survival. Patients’ age, pulmonary dysfunction and transfer from an external institution were predictors for ECMO therapy. The interaction of patient age with D-dimer levels on admission and creatinine levels with SOFA score without GCS were predictors for renal replacement therapy.ConclusionsUsing Explainable Boosting Machine analysis, we confirmed and weighed previously reported and identified novel predictors for outcome in critically ill COVID-19 patients. Using this strategy, predictive modeling of COVID-19 ICU patient outcomes can be performed overcoming the limitations of linear regression models.Trial registration “ClinicalTrials” (clinicaltrials.gov) under NCT04455451.

Nonlinear compressed sensing-based adaptive modal shapes selection approach for efficient dynamic response analysis of flexible multibody system

Nonlinear compressed sensing-based adaptive modal shapes selection approach for efficient dynamic response analysis of flexible multibody system

Weakly nonlinear shape oscillations of inviscid drops

Abstract

Non-linear association of anthropometric measurements and pulmonary function

This study examined the association of anthropometric measurements [body mass index (BMI), waist circumference (WC), percentage body fat (PBF), body roundness index (BRI) and A Body Shape Index (ABSI)] with pulmonary function using a United States national cohort. This cross-sectional study included 7346 participants. The association between anthropometric measurements and pulmonary function was assessed by multivariable linear regression. Where there was evidence of non-linearity, we applied a restricted cubic spline to explore the non-linear association. All analyses were weighted to represent the U.S. population and to account for the intricate survey design. After adjusting for age, race, education, smoking, and physical activity, both underweight and obesity were associated with reduced forced expiratory volume in 1 s (FEV1) and forced vital capacity (FVC). Furthermore, the associations between BMI and FEV1, as well as FVC, were reversed U-shape in both males and females. Similar non-linear association shape occurred in WC, PBF, BRI and ABSI. Conclusion: BMI, WC, PBF, BRI, ABSI are non-linearly associated with pulmonary function. Reduced pulmonary function is a risk factor for future all-cause mortality and cardiovascular events; thus, this nonlinearity may explain the U-shape or J-shape association of BMI with overall mortality and cardiovascular events.

Rigid motion invariant statistical shape modeling based on discrete fundamental forms: Data from the osteoarthritis initiative and the Alzheimer’s disease neuroimaging initiative

We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer’s disease and osteoarthritis, respectively. In particular, we outperform state-of-the-art classifiers based on geometric deep learning as well as statistical shape modeling especially in presence of sparse training data. To provide insight into the model’s ability of capturing biological shape variability, we carry out an analysis of specificity and generalization ability.

Large Amplitude Forced Vibrations of Restrained Beams Resting on Elastic Point Supports

The present paper concerns the study of geometrically non-linear forced vibrations of beams resting on two different types of springs: rotational and translational. Assuming that the motion is harmonic, the displacement is extended as a series of spatial functions determined by solving the linear problem. Hamilton’s principle and spectral analysis are used to reduce the problem to a non-linear algebraic system solved using a previously developed approximate method. The effects of the nature of the added springs and their location on the non-linear behaviour of the beam are examined. A multimode approach is used in the forced case to obtain results over a wide range of vibration amplitudes. This leads to examining the non-linear forced dynamic response for different positions of each spring and different levels of excitations. Following a parametric study, the non-linear forced mode shapes and their associated bending moments are presented for different levels of excitations and for different vibration amplitudes to give an estimation of the stress distribution over the beam length.

Revisiting the link between extreme sea levels and climate variability using a spline-based non-stationary extreme value analysis

Non-stationary extreme value analysis is a powerful framework to address the problem of time evolution of extremes and its link to climate variability as measured by different climate indices CI (like North Atlantic Oscillation NAO index). To model extreme sea levels (ESLs), a widely-used tool is the non-stationary Generalized Extreme Value distribution (GEV) where the parameters (location, scale and shape) are allowed to vary as a function of some covariates like the month-of-year or some CIs. A commonly used assumption is that only a few CIs impact the GEV parameters by using a linear model, and most of the time by focusing on two GEV parameters (location or/and the scale parameter). In the present study, these assumptions are revisited by relying on a data-driven spline-based GEV fitting approach combined with a penalization procedure. This allows identifying the type (non- or linear) of the CI influence for any of the three GEV parameters directly from the data, and evaluating the significance of this relation, i.e. without making any a priori assumptions as it is traditionally done. This approach is applied to the monthly maxima of sea levels derived from eight of the longest (quasi century-long) tide gauge dataset (Brest, France; Cuxhaven, Germany; Gedser, Denmark; Halifax, Canada; Honolulu, US; Newlyn, UK; San Francisco, US; Stockholm, Sweden) and by accounting for four major CIs (the North Atlantic Oscillation, the Atlantic Multidecadal Oscillation, the Niño 1 + 2 and the Southern Oscillation indices). From this analysis, we show that: (1) the links between CIs and different parameters of a GEV distribution fitted to ESL data are most of the time linear, but some of them present significant non-linear shapes; (2) multiple CIs should be considered to predict ESLs, and (3) the CI influence of the GEV distribution is not limited to the location parameter. These results are useful to understand current modes of variability of ESLs, and ultimately to improve coastal resilience through more precise extreme water level assessments.

Do foreign direct investment and environmental regulation improve green technology innovation? An empirical analysis based on panel data from the Chinese manufacturing industry.

The environmental regulation and foreign direct investment (FDI) inflow have an important impact on the progress of green technology. This study analyzes the impacts of environmental regulation and FDI on green technology innovation (GTI) based on the panel data of 13 Chinese manufacturing sectors. The results of static panel regression show that the environmental regulation has a positive impact on GTI, while the FDI has a negative impact. The results of the panel threshold model reveal that the effect of environmental regulation on GTI presents a nonlinear shape. The negative effect of FDI on GTI is strengthened when the environmental regulation exceeds its threshold. Increasing FDI inflow can inhibit the effect of environmental regulation. Meanwhile, a strict environmental regulation can enhance the inhibiting effect of FDI on GTI. The FDI inflow into high-tech manufacturing sectors has a less negative impact on GTI than the FDI inflow into low-tech sectors in the case of the enhancement of environmental regulation. This study provides some implications for the formulation of environmental regulation and the FDI inflow into China to improve the GTI.

On the Generation of Harmonics by the Non-Linear Buckling of an Elastic Beam

The Euler–Bernoulli theory of beams is usually presented in two forms: (i) in the linear case of a small slope using Cartesian coordinates along and normal to the straight undeflected position; and (ii) in the non-linear case of a large slope using curvilinear coordinates along the deflected position, namely, the arc length and angle of inclination. The present paper starts with the exact equation in a third form, that is, (iii) using Cartesian coordinates along and normal to the undeflected position like (i), but allowing exactly the non-linear effects of a large slope like (ii). This third form of the equation of the elastica shows that the exact non-linear shape is a superposition of linear harmonics; thus, the non-linear effects of a large slope are equivalent to the generation of harmonics of a linear solution for a small slope. In conclusion, it is shown that: (i) the critical buckling load is the same in the linear and non-linear cases because it is determined by the fundamental mode; (ii) the buckled shape of the elastica is different in the linear and non-linear cases because non-linearity adds harmonics to the fundamental mode. The non-linear shape of the elastica, for cases when powers of the slope cannot be neglected, is illustrated for the first four buckling modes of cantilever, pinned, and clamped beams with different lengths and amplitudes.

Nonlinear Normal Mode backbone estimation with near-resonant steady state inputs

This work presents a new technique for nonlinear system identification that utilizes near-resonant steady-state harmonically excited vibration measurements to estimate the Nonlinear Normal Mode backbones. The algorithm is based on the previously proposed Single Nonlinear Resonant Mode formula and uses it in a new and more effective way to estimate one point on the nonlinear mode from only one steady-state measurement collected near the resonance. A by-product of this work is a derivation of a novel formula expressing how the damping ratio changes with the motion amplitude. Several measurements at various forcing amplitudes can be combined to estimate the nonlinear mode and damping as a function of amplitude, which can be further used to predict the forced steady-state response of the structure in the vicinity of the mode of interest. Compared to existing phase resonance methods, the proposed technique can reduce the time required to obtain measurements and avoids difficulties due to e.g. the premature jump phenomenon. The algorithm assumes that the modes are well-separated and no internal resonances are present in the system. Additionally, it requires the accurate identification of linear modes in the low-amplitude vibration tests and it assumes that the nonlinear normal mode shape does not change significantly with response amplitude The method is first evaluated numerically using reduced models of clamped–clamped flat and curved beams that exhibit both stiffening and softening–stiffening responses, respectively. Then the method is employed experimentally to measure the NNM backbones of beams that were manufactured from polylactide using a 3D printer and experience significant eigen-frequency shifts when the motion amplitude increases. The results are validated against measurements collected using the traditional phase resonance testing approach.

Geometrical nonlinearities and shape effects in electromechanical models of piezoelectric bridge structures

We consider nonlinear shape effects appearing in the lumped electromechanical model of a bimorph piezoelectric bridge structure due to the interaction between the electromechanical constitutive model and the geometry of the structure. At finite proof-mass displacement and electrode voltage, the shape of the beams is no longer given by Euler-Bernoulli theory which implies that shape effects enter in both the electrical and mechanical domains and in the coupling between them. Accounting for such effects is important for the accurate modelling of, e.g., piezoelectrical energy harvesters and actuators in the regime of large deflections and voltages. We present a general method, based on a variational approach minimizing the Gibbs enthalpy of the system, for computing corrections to the nominal shape function and the associated corrections to the lumped model. The lowest order correction is derived explicitly and is shown to produce significant improvements in model accuracy, both in terms of the Gibbs enthalpy and the shape function itself, over a large range of displacements and voltages. Furthermore, we validate the theoretical model using large deflection finite element simulations of the bridge structure and conclude that the lowest order correction substantially improve the model, obtaining a level of accuracy expected to be sufficient for most applications. Finally, we derive the equations of motion for the lowest order corrected model and show how the coupling between the electromechanical properties and the geometry of the bridge structure introduces nonlinear interaction terms.

DATA REDUCTION BY THE HAAR FUNCTION: A CASE STUDY OF THE PHILLIPS CURVE

The unorthodox estimation procedure, which Phillips (1958) adopted in his original paper, is examined using the Haar wavelet filter. The application of the Haar wavelet transform to Phillips’ original data shows that Phillips’ six pairs of mean coordinates display a striking similarity with the Haar scaling coefficients that represent averages with a period greater than 16 years. This is consistent with Desai’s (1975) intuition on the interpretation of the Phillips Curve. We show that the choice of sorting observations by ascending values of the unemployment rate is crucial for reaching the goal of estimating the eye-catching nonlinear hyperbolic shape of the wage–unemployment relationship that would be otherwise linear. Interestingly, the Haar filter can account not only for the facts characterizing the Phillips’ relationship up to the early 1960s but also for two important facts mostly debated among policymakers: the downward shift of the Phillips Curve and its flattening over time.

Solving nonlinear third-order boundary value problems based-on boundary shape functions

Abstract For a third-order nonlinear boundary value problem (BVP), we develop two novel methods to find the solutions, satisfying boundary conditions automatically. A boundary shape function (BSF) is created to automatically satisfy the boundary conditions, which is then employed to develop new numerical algorithms by adopting two different roles of the free function in the BSF. In the first type algorithm, we let the BSF be the solution of the BVP and the free function be a new variable. In doing so, the nonlinear BVP is certainly and exactly transformed to an initial value problem for the new variable with its terminal values as unknown parameters, whereas the initial conditions are given. In the second type algorithm, let the free functions be a set of complete basis functions and the corresponding boundary shape functions be the new bases. Since the solution already satisfies the boundary conditions automatically, we can apply a simple collocation technique inside the domain to determine the expansion coefficients and then the solution is obtained. For the general higher-order boundary conditions, the BSF method (BSFM) can easily and quickly find a very accurate solution. Resorting on the BSFM, the existence of solution is proved, under the Lipschitz condition for the ordinary differential equation system of the new variable. Numerical examples, including the singularly perturbed ones, confirm the high performance of the BSF-based numerical algorithms.

A Numerical Study of Wave‐Driven Mean Flows and Setup Dynamics at a Coral Reef‐Lagoon System

AbstractTwo‐dimensional mean wave‐driven flow and setup dynamics were investigated at a reef‐lagoon system at Ningaloo Reef, Western Australia, using the numerical wave‐flow model, SWASH. Phase‐resolved numerical simulations of the wave and flow fields, validated with highly detailed field observations (including >10 sensors through the energetic surf zone), were used to quantify the main mechanisms that govern the mean momentum balances and resulting mean current and setup patterns, with particular attention to the role of nonlinear wave shapes. Momentum balances from the phase‐resolved model indicated that onshore flows near the reef crest were primarily driven by the wave force (dominated by radiation stress gradients) due to intense breaking, whereas the flow over the reef flat and inside the lagoon and channels was primarily driven by a pressure gradient. Wave setup inside the lagoon was primarily controlled by the wave force and bottom stress. The bottom stress reduced the setup on the reef flat and inside the lagoon. Excluding the bottom stress contribution in the setup balance resulted in an over prediction of the wave‐setup inside the lagoon by up to 200–370%. The bottom stress was found to be caused by the combined presence of onshore directed wave‐driven currents and (nonlinear) waves. Exclusion of the bottom stress contribution from nonlinear wave shapes led to an over prediction of the setup inside the lagoon by approximately 20–40%. The inclusion of the nonlinear wave shape contribution to the bottom stress term was found to be particularly relevant in reef regions that experience a net onshore mass flux over the reef crest.

Optimization of the Opening Shape in Slot Spray Nozzles in a Field Boom Sprayer

The European Standard ISO 16122-2:2015 requires that the coefficient of variation for sprayed liquid fall should not exceed 10%. Field sprayers generate a stream of liquid sprayed at an angle that depends on the design of the nozzles. Among field boom sprayers, previous methods for selecting the shape of the opening of a single-slotted spray nozzle have been based on the following rectangular, triangular, normal, beta, and truncated normal distributions; distribution obtained from a nozzle with a stream in the form of an empty cone; and glued square functions. These methods, however, have provided a limited range of uniformity. Consequently, the initial assumption that the monotonicity of the function corresponds to the shape of a quarter of the symmetrical oval nozzle opening allows for a full computerized optimization of nozzle shape with a spray angle of α = 110° (or α = 120°). In this case, the spray uniformity parameter is controlled and freely declines almost to zero. In this study, based on the nonlinear shape obtained, we developed the shape of the nozzle outlet opening with a coefficient of variation of 0.388% using spline linear functions. Further applications of the symmetry of the developed model would allow for multiple modifications of the shape of this opening, and therefore, without changing the spray uniformity parameter, nozzles with slightly different characteristics could be obtained.