Manifold Approach Research Articles

Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach

A comparison between two methods to derive reduced-order models (ROM) for geometrically nonlinear structures is proposed. The implicit condensation and expansion (ICE) method relies on a series of applied static loadings. From this set, a stress manifold is constructed for building the ROM. On the other hand, nonlinear normal modes rely on invariant manifold theory in order to keep the key property of invariance for the reduced subspaces. When the model coefficients are fully known, the ICE method reduces to a static condensation. However, in the framework of finite element discretization, getting all these coefficients is generally too computationally expensive. The stress manifold is shown to tend to the invariant manifold only when a slow/fast decomposition between master and slave coordinates can be assumed. Another key problem in using the ICE method is related to the fitting procedure when a large number of modes need to be taken into account. A simplified procedure, relying on normal form theory and identification of only resonant monomial terms in the nonlinear stiffness, is proposed and contrasted with the current method. All the findings are illustrated on beams and plates examples.

On Solutions of the Nonlinear Heat Equation Using Modified Lie Symmetries and Differentiable Topological Manifolds

In this paper, we present three simple analytical techniques for obtaining solutions of the nonlinear heat equations. The heat equations, both linear and nonlinear, are very important to the mathematical sciences. This is because they are reduced forms of many models, hard to solve directly. The techniques are based on Lie’ symmetry group theoretical methods. The first is the pure Lie approach, followed by our modified Lie approach. The third is our differentiable topological manifolds approach. As an application, we determine the separation distance, in the quantum superposition principle, relevant to nanoscience.

Correction To: Existence Results for Fractional p(x, .)-Laplacian Problem Via the Nehari Manifold Approach

Correction To: Existence Results for Fractional p(x, .)-Laplacian Problem Via the Nehari Manifold Approach

Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

Nehari manifold approach for fractional p(.)-Laplacian system involving concave-convex nonlinearities

In this article, using Nehari manifold method we study the multiplicity of solutions of the nonlocal elliptic system involving variable exponents and concave-convex nonlinearities, $$\displaylines{ (-\Delta)_{p(\cdot)}^{s} u=\lambda a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \quad x\in \Omega; \cr (-\Delta)_{p(\cdot)}^{s} v=\mu b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\quad x\in \Omega; \cr u=v=0,\quad x\in \Omega^c:=\mathbb R^N\setminus\Omega, }$$ where \(\Omega\subset\mathbb R^N\), \(N\geq2\) is a smooth bounded domain, \(\lambda,\mu>0\) are parameters, and \(s\in(0,1)\). We show that there exists \(\Lambda>0\) such that for all \(\lambda+\mu<\Lambda\), this system admits at least two non-trivial and non-negative solutions under some assumptions on \(q,\alpha,\beta,a,b,c\).
 For more information see https://ejde.math.txstate.edu/Volumes/2020/98/abstr.html

Load Magnitude and Locomotion Pattern Alter Locomotor System Function in Healthy Young Adult Women.

IntroductionDuring cyclical steady state ambulation, such as walking, variability in stride intervals can indicate the state of the system. In order to define locomotor system function, observed variability in motor patterns, stride regulation and gait complexity must be assessed in the presence of a perturbation. Common perturbations, especially for military populations, are load carriage and an imposed locomotion pattern known as forced marching (FM). We examined the interactive effects of load magnitude and locomotion pattern on motor variability, stride regulation and gait complexity during bipedal ambulation in recruit-aged females.MethodsEleven healthy physically active females (18–30 years) completed 1-min trials of running and FM at three load conditions: no additional weight/bodyweight (BW), an additional 25% of BW (BW + 25%), and an additional 45% of BW (BW + 45%). A goal equivalent manifold (GEM) approach was used to assess motor variability yielding relative variability (RV; ratio of “good” to “bad” variability) and detrended fluctuation analysis (DFA) to determine gait complexity on stride length (SL) and stride time (ST) parameters. DFA was also used on GEM outcomes to calculate stride regulation.ResultsThere was a main effect of load (p = 0.01) on RV; as load increased, RV decreased. There was a main effect of locomotion (p = 0.01), with FM exhibiting greater RV than running. Strides were regulated more tightly and corrected quicker at BW + 45% compared (p < 0.05) to BW. Stride regulation was greater for FM compared to running. There was a main effect of load for gait complexity (p = 0.002); as load increased gait complexity decreased, likewise FM had less (p = 0.02) gait complexity than running.DiscussionThis study is the first to employ a GEM approach and a complexity analysis to gait tasks under load carriage. Reduction in “good” variability as load increases potentially exposes anatomical structures to repetitive site-specific loading. Furthermore, load carriage magnitudes of BW + 45% potentially destabilize the system making individuals less adaptable to additional perturbations. This is further evidenced by the decrease in gait complexity, which all participants demonstrated values similarly observed in neurologically impaired populations during the BW + 45% load condition.

Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of the phase space. The second method is the modal derivative approach, and more specifically, the quadratic manifold defined in order to derive a second-order nonlinear change of coordinates. Both methods share a common point of view, willing to introduce a nonlinear mapping to better define a reduced-order model that could take more properly into account the nonlinear restoring forces. However, the calculation methods are different and the quadratic manifold approach has not the invariance property embedded in its definition. Modal derivatives and static modal derivatives are investigated, and their distinctive features in the treatment of the quadratic nonlinearity are underlined. Assuming a slow/fast decomposition allows understanding how the three methods tend to share equivalent properties. While they give proper estimations for flat symmetric structures having a specific shape of nonlinearities and a clear slow/fast decomposition between flexural and in-plane modes, the treatment of the quadratic nonlinearity makes the predictions different in the case of curved structures such as arches and shells. In the more general case, normal form approach appears preferable since it allows correct predictions of a number of important nonlinear features, including the hardening/softening behaviour, whatever the relationships between slave and master coordinates are.

In-Situ Adaptive Manifolds: Enabling computationally efficient simulations of complex turbulent reacting flows

Reduced-order manifold approaches to turbulent combustion modeling traditionally involve precomputation of manifold solutions and pretabulation of the thermochemical database versus a small number of manifold variables. However, additional manifold variables are required as the complexity of turbulent combustion processes increases through consideration of, for example, multi-modal, non-adiabatic, or non-isobaric combustion, or combustion featuring multiple and/or inhomogeneous inlets. This increase in the number of manifold variables comes with an increase in the computational cost of precomputing a greater number of manifold solutions, most of which are never actually utilized in a CFD calculation. The memory required to store the pretabulated high-dimensional thermochemical database also increases, practically limiting the complexity of manifold-based combustion models. In this work, a new In-Situ Adaptive Manifolds (ISAM) approach is developed that overcomes this limitation by combining ‘on-the-fly’ calculation of manifold solutions with In-Situ Adaptive Tabulation (ISAT), enabling the use of more complex manifold-based turbulent combustion models. The performance of ISAM is evaluated via LES of turbulent nonpremixed jet flames with both hydrogen and hydrocarbon fuels. A performance assessment indicates that the computational overhead associated with ISAM compared to pretabulation ranges from negligible up to a factor of two, with most of this overhead associated with convolution of the thermochemical state against a presumed subfilter PDF. In addition, the memory requirements of ISAM are more than two orders of magnitude less than conventional tabulation. These results demonstrate the potential for ISAM to accommodate significantly more complex manifold-based combustion models.

Large-Eddy Simulation of a Model Aero-Engine Sooting Flame With a Multiphysics Approach

In this work, an in-house developed multiphysics unsteady approach (U-THERM3D) is employed to compute wall temperatures in a loosely coupled manner for a model aero-engine combustor, fueled with a non-premixed ethylene/air mixture and representative of the RQL technology. Large-Eddy Simulation (LES) is carried out to prove that scale-resolving approaches are required for a reliable prediction of turbulent mixing with respect to steady-state methods. A Flamelet Generated Manifold approach is adopted to describe the reactive flow, whereas two additional transport equations are included to predict the soot concentration. For a thorough investigation of the considered aero-engine combustor, cold flow results are firstly presented, highlighting the importance of an accurate predictions of general flow features. Then, the aerothermal fields are analysed to show the overall accuracy of the developed approach and so the impact of heat losses, comparing the results of an adiabatic simulation with respect to the loosely coupled conjugate heat transfer calculation. Finally, the investigation is focused on the prediction of the wall temperature distribution to prove the advantages of the U-THERM3D procedure with respect to steady-state approach.

Ground state solutions for quasilinear Schrodinger equations with periodic potential

This article concerns the quasilinear Schrodinger equation $$\displaylines{ -\Delta u-u\Delta (u^2)+V(x)u=K(x)|u|^{2\cdot2^*-2}u+g(x,u),\quad x\in\mathbb{R}^N, \cr u\in H^1(\mathbb{R}^N),\quad u&gt;0, }$$ where V and K are positive, continuous and periodic functions, g(x,u) is periodic in x and has subcritical growth. We use the generalized Nehari manifold approach developed by Szulkin and Weth to study the ground state solution, i.e. the nontrivial solution with least possible energy.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/82/abstr.html

Assessment of the impact of subgrid-scale stress models and mesh resolution on the LES of a partially-premixed swirling flame

Large Eddy Simulation (LES) has become an attractive option for the modelling of turbulent combustion, and a significant effort has been put into the development of suitable models for combustion at the subgrid-scale (SGS). Comparatively, little attention has been paid to the impact of SGS stress models in the simulation of complex reactive flows. In the present work, a partially-premixed case from the Sydney swirl-stabilised flames series has been investigated using LES and the Flamelet Generated Manifold (FGM) approach, employing four different SGS stress models on three numerical grids with different levels of refinement, quantified with an a priori mesh quality criterion. It was found that the use of advanced SGS models can improve the accuracy of the results significantly in both the highly turbulent region close to the burner and in the low-turbulence region further downstream. Also, a physically sound formulation for SGS stresses was found to reduce the sensitivity of the results to mesh refinement, giving acceptable results on relatively coarse grids. This can be a significant advantage in the modelling of complex geometries where there is a need to find a compromise between mesh refinement and computational costs.

Existence and multiplicity of solutions for singular fractional elliptic system via the Nehari manifold approach

Existence and multiplicity of solutions for singular fractional elliptic system via the Nehari manifold approach

Learning Common Harmonic Waves on Stiefel Manifold - A New Mathematical Approach for Brain Network Analyses

Converging evidence shows that disease-relevant brain alterations do not appear in random brain locations, instead, its spatial pattern follows large scale brain networks. In this context, a powerful network analysis approach with a mathematical foundation is indispensable to understand the mechanism of neuropathological events spreading throughout the brain. Indeed, the topology of each brain network is governed by its native harmonic waves, which are a set of orthogonal bases derived from the Eigen-system of the underlying Laplacian matrix. To that end, we propose a novel connectome harmonic analysis framework to provide enhanced mathematical insights by detecting frequency-based alterations relevant to brain disorders. The backbone of our framework is a novel manifold algebra appropriate for inference across harmonic waves that overcomes the limitations of using classic Euclidean operations on irregular data structures. The individual harmonic difference is measured by a set of common harmonic waves learned from a population of individual Eigen systems, where each native Eigen-system is regarded as a sample drawn from the Stiefel manifold. Specifically, a manifold optimization scheme is tailored to find the common harmonic waves which reside at the center of Stiefel manifold. To that end, the common harmonic waves constitute the new neuro-biological bases to understand disease progression. Each harmonic wave exhibits a unique propagation pattern of neuro-pathological burdens spreading across brain networks. The statistical power of our novel connectome harmonic analysis approach is evaluated by identifying frequency-based alterations relevant to Alzheimer's disease, where our learning-based manifold approach discovers more significant and reproducible network dysfunction patterns compared to Euclidian methods.

Large eddy simulation of a turbulent non-premixed flame based on the flamelet-generated manifolds approach and a reduced mechanism verification

This work presents the design of a large eddy simulation based on the flamelet-generated manifolds (FGM) approach to evaluate a Sandia Flame D, which is a turbulent non-premixed piloted jet flame. The detailed kinetic mechanism of the Gas Research Institute (GRI3.0, which contains 53 species and 325 elementary reactions), as well as a reduced mechanism, are employed to describe combustion. This reduced mechanism, containing 19 species and 68 elementary reactions, is proposed based on the direct relation graphs with error-propagation (DRGEP) method and a species sensitivity analysis from the GRI3.0. Both the two mechanisms are introduced into Fluent to simulate a Sandia Flame D, and the numerical results are in good agreement with the experimental data, which indicates the accurate predictive capability of the LES/FGM method in combustion modeling and that the proposed reduced mechanism can be applied to the numerical simulation in practical engineering problems.

Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth

The aim of this paper is to study a degenerate Kirchhoff-type elliptic problem driven by the fractional Laplace operator with variable order derivative and variable exponents. More precisely, we consider{[u]s(⋅)2(θ−1)(−Δ)s(⋅)u=λa(x)|u|p(x)−2u+b(x)|u|q(x)−2uin Ω,u=0in RN∖Ω, where [u]s(⋅) is the Garliado seminorm, θ>1, N≥1, s(⋅):RN×RN→(0,1) is a continuous function, λ>0 is a parameter, Ω is a bounded domain in RN with N>2s(x,y) for all (x,y)∈Ω×Ω, (−Δ)s(⋅) is the variable-order fractional Laplacian, a,b∈L∞(Ω) are two positive weight functions, and p,q∈C(Ω‾) with 1<p(x)<2θ<q(x)<2N/(N−2s(x,x)). Under some suitable assumptions, we obtain that the problem admits at least two distinct nonnegative solutions by applying the Nehari manifold approach, provided λ is sufficiently small. Moreover, the existence of infinitely many solutions is also investigated by the symmetric mountain pass theorem.

A damage-based numerical manifold approach to crack propagation in rocks

A damaged-based numerical manifold method is proposed in this paper to investigate the micro- and macro- mechanical properties of rocks. This model accounts for material heterogeneity and local material degradation using an exponential material softening law. The Mohr-Coulomb criterion with a tension cut-off is chosen as the strength criterion to capture the rock damage and fracture modes. Once the induced tensile or compressive stress in an element fulfills the criterion, this element will be damaged and a crack will initiate from the centroid of this damaged element to the centroid of its adjacent damaged manifold element. To verify the proposed damage-based numerical manifold method (NMM), a series of benchmark tests are conducted and the results are compared with the numerical manifold method incorporated with a Mohr-Coulomb criterion-based fracturing algorithm to simulate the progressive failure. The simulated results are consistent with the analytical solutions, and the advantages and disadvantages of two methods are compared and analyzed. Numerical results indicate that the proposed damage-based numerical manifold method can capture the process of crack initiation and propagation, and this method is an effective and reliable approach for modelling cracking behavior of rocks.

Existence of solutions for a class of singular quasilinear elliptic equations

ABSTRACT In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems where , the potential and . By Nehari manifold approach and the technique of splitting a minimizing sequence, we prove the existence of positive solutions for the above problem.

Existence Results for Fractional p(x, .)-Laplacian Problem Via the Nehari Manifold Approach

In the present paper, we study the existence of two non-negative solutions for a class of fractional p(x, .)-Laplacian problems with non-negative weight functions. The main tool is the Nehari manifold approach. Moreover, under some suitable assumptions, continuous and compact embeddings results are established.

Chern-Simons supergravity on supergroup manifolds

We construct N=1 d=3 AdS supergravity within the group manifold approach and compare it with Achucarro-Townsend Chern-Simons formulation of the same theory. We clarify the relation between the off-shell super gauge transformations of the Chern- Simons theory and the off-shell worldvolume supersymmetry transformations of the group manifold action. We formulate the Achucarro-Townsend model in a double supersymmetric action where the Chern-Simons theory with a supergroup gauge symmetry is constructed on a supergroup manifold. This framework is useful to establish a correspondence of degrees of freedom and auxiliary fields between the two descriptions of d=3 supergravity.

Migrant family entrepreneurship – mixed and multiple embeddedness of transgenerational Turkish family entrepreneurs in Berlin

PurposeBy combining manifold approaches from migrant entrepreneurship and family business studies, the purpose of the paper is to shed some light upon the contextual features of motivation, resources, generational pathways of Turkish migrant family entrepreneurs in Berlin – through the lens of a mixed and multiple embeddedness approach.Design/methodology/approachAn explorative research design, based on an eclectic theoretical framework and on purposive sampling, combines qualitative in-depth interviews/content analysis and on-site observation resulting in an almost ethnographic assessment of selected case studies of Turkish migrant family entrepreneurs (concerning age (min. 20 years), size (15+ employees) and currently at a stage of succession).FindingsThe results show that despite specific strategies vary – four circumstances hold true for all cases: (1) firm trajectories were characterized by little strategic planning and mostly trail-and error processes in the past and business survival is highly dependent on owner families; (2) owner families heavily relied on personal, family and collective resources, not benefiting from promotion programmes or micro-funding measures for SMEs; (3) owner families have actively developed their (mixed) embeddings during the growth of their migrant business beyond the single ethnic group at various spatial scales; (4) succession adds another layer of context – what we call here multiple embeddedness – with ambivalent effects: emerging potentials and conflicts between the preceding and succeeding generation.Practical implicationsResults have shown that is it necessary to set up both: customized funding opportunities for migrant start-ups in general and succession consulting for migrant family entrepreneurs in particular. Given the magnitude of family migrant entrepreneurs and the accelerating migration patterns in most Western European countries, there is urgent need for such measures.Originality/valueFamily entrepreneurship has been often discussed without a migration perspective, neither taking a systematic look at pertinent motivation, resources, and future trajectories nor context. Migrant entrepreneurship studies barely take the family or family-specific issues (e.g. succession) into account, and mainly deal with the integration or economic aspects. Our mixed and multiple embeddedness approach allows for a holistic view on transgenerational migrant family entrepreneurship by integrating both socio-spatial (actor, family, network, micro, meso, macro) and multi-generational contexts (preceding, succeeding).