Implicit Function Research Articles

HYSTERESIS OF IMPLICIT EQUATIONS IN HYPOPLASTICITY FOR SOIL MATERIALS WITH GRANULAR HARDNESS DEGRADATION

AbstractWe study a hypoplastic model for soil and granular materials stemming from geomechanical engineering which further incorporates effects of degradation of the granular hardness, therefore allowing for the description of environmental weathering. The governing system is described by a nonlinear system of transcendental-differential equations for stress and strain rate, which is investigated with respect to its long-time dynamic. Under deviatoric stress control, two different solutions of the underlying, implicit differential equations are constructed analytically. The spherical components of stress and strain rate converge asymptotically to an attractor and lead to the sparsification of material states. Whereas under cyclic loading-unloading carried out in a numerical simulation, finite ratcheting of the deviatoric strain rate is observed in the form of a square spiral.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Bending of Lloyd’s mirror to eliminate the period chirp in the fabrication of diffraction gratings

We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.

A double-loop adaptive relevant vector machine combined with Harris Hawks optimization-based importance sampling

PurposeEnsuring the safety of structures is important. However, when a structure possesses both an implicit performance function and an extremely small failure probability, traditional methods struggle to conduct a reliability analysis. Therefore, this paper proposes a reliability analysis method aimed at enhancing the efficiency of rare event analysis, using the widely recognized Relevant Vector Machine (RVM).Design/methodology/approachDrawing from the principles of importance sampling (IS), this paper employs Harris Hawks Optimization (HHO) to ascertain the optimal design point. This approach not only guarantees precision but also facilitates the RVM in approximating the limit state surface. When the U learning function, designed for Kriging, is applied to RVM, it results in sample clustering in the design of experiment (DoE). Therefore, this paper proposes a FU learning function, which is more suitable for RVM.FindingsThree numerical examples and two engineering problem demonstrate the effectiveness of the proposed method.Originality/valueBy employing the HHO algorithm, this paper innovatively applies RVM in IS reliability analysis, proposing a novel method termed RVM-HIS. The RVM-HIS demonstrates exceptional computational efficiency, making it eminently suitable for rare events reliability analysis with implicit performance function. Moreover, the computational efficiency of RVM-HIS has been significantly enhanced through the improvement of the U learning function.

Modeling of implicit multi term fractional delay differential equation: Application in pollutant dispersion problem

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances.

Learning continuous shape priors from sparse data with neural implicit functions.

Statistical shape models are an essential tool for various tasks in medical image analysis, including shape generation, reconstruction and classification. Shape models are learned from a population of example shapes, which are typically obtained through segmentation of volumetric medical images. In clinical practice, highly anisotropic volumetric scans with large slice distances are prevalent, e.g., to reduce radiation exposure in CT or image acquisition time in MR imaging. For existing shape modeling approaches, the resolution of the emerging model is limited to the resolution of the training shapes. Therefore, any missing information between slices prohibits existing methods from learning a high-resolution shape prior. We propose a novel shape modeling approach that can be trained on sparse, binary segmentation masks with large slice distances. This is achieved through employing continuous shape representations based on neural implicit functions. After training, our model can reconstruct shapes from various sparse inputs at high target resolutions beyond the resolution of individual training examples. We successfully reconstruct high-resolution shapes from as few as three orthogonal slices. Furthermore, our shape model allows us to embed various sparse segmentation masks into a common, low-dimensional latent space - independent of the acquisition direction, resolution, spacing, and field of view. We show that the emerging latent representation discriminates between healthy and pathological shapes, even when provided with sparse segmentation masks. Lastly, we qualitatively demonstrate that the emerging latent space is smooth and captures characteristic modes of shape variation. We evaluate our shape model on two anatomical structures: the lumbar vertebra and the distal femur, both from publicly available datasets.

Continuous/smooth transonic solutions to the quasi-one-dimensional steady relativistic Euler equations

This paper establishes the existence and uniqueness of continuous/smooth Meyer type transonic solutions to the quasi-one-dimensional steady isentropic relativistic Euler equations. We use the high-order Taylor’s expansion of the velocity near the sonic point to overcome the difficulties of applying the implicit function theorem caused by the sonic degeneracy. It is verified that the flow in the De Laval nozzle accelerates from subsonic upstream and continuously/smoothly pass through the sonic surface at the throat and then becomes supersonic downstream. Moreover, the flow may have a zero or positive acceleration at the sonic surface, which depends on the geometric property of the nozzle at the throat.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers–Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

A novel lattice structure design approach based on Schwarz Primitive triply periodic minimal surfaces

In recent years, lattice structures based on triply periodic minimal surfaces have attracted the attention of researchers worldwide due to their exceptional geometrical and mechanical features. In this paper, using two distinct implicit functions for the rotation angle and the axis of rotation, the surface points of the Schwarz’ Primitive cellular lattice are moved to a new position to construct some novel lattices. Various cellular lattices are then generated by manipulating different design parameters and investigated using finite element method to evaluate porosity, surface-to-volume ratio, elastic modulus and Zener ratio. The findings indicate that although the porosity doesn’t change profoundly by applying the transformation, the surface-to-volume ratio and elastic modulus increases and decreases respectively as the maximum rotation angle increases. In addition, Zener ratio exhibits non-linear variation with the transformation, potentially increasing or decreasing by increasing the maximum rotation angle, depending on other parameters. The maximum difference between the values of surface-to-volume ratio, elastic modulus, and Zener ratio of the novel lattices and those of the original one is 16.9% (for one case it decreases by 68.7%), 68.5%, and 45.6%, respectively. These observations suggest that the proposed method might presents significant potential for facilitating the creation of innovative shell-based lattice structures.

Multimodal surface wave inversion with automatic differentiation

Summary Investigating subsurface shear-wave velocity (vs) structures using surface wave dispersion data involves minimizing a misfit function that is commonly solved through gradient-based optimization. Sensitivity kernels for model updates are commonly estimated using numerical differentiation, variational methods or implicit functions which however, may involve numerical instability and computational challenges when dealing with complex velocity models and large datasets. In this study, we propose a novel surface wave inversion framework in which error-free gradients are calculated by automatic differentiation (AD) and forward modeling is implemented by convenient computational graphs in the state-of-the-art deep learning framework. The AD-based inversion approach is first validated using two synthetic datasets. Then the subsurface structures at three distinct locations, namely the Great Plains and the Long Beach in the US and Tong Zhou in China, are also derived using this method with seismic ambient noise data, which show nice consistency with those obtained using traditional methods. With the significantly improved computational efficiency, a great number of initial models can be inverted simultaneously to mitigate the impact of local minima and to estimate the uncertainty in the invert models. We have developed a new surface wave inversion package named ADsurf based on automatic differentiation and computational graphs in the deep learning framework, and its computational efficiency is also compared with the traditional finite-difference based gradient estimation approach. While a great number of intriguing studies on the geophysical inverse problems have been conducted recently using deep learning for end-to-end mapping, the use of AD provided in the in the deep learning frameworks to assist and expedite the gradient computations are still underexploited in geophysics. Thus, it is expected that various geophysical inverse problems in many different areas beyond the surface wave inversion can also be tackled with this new paradigm in the future.

On the computation of the gradient in implicit neural networks

AbstractImplicit neural networks and the related deep equilibrium models are investigated. To train these networks, the gradient of the corresponding loss function should be computed. Bypassing the implicit function theorem, we develop an explicit representation of this quantity, which leads to an easily accessible computational algorithm. The theoretical findings are also supported by numerical simulations.

Semi-analytical solutions for forced and free vibration of damped fluid-conveying pipe systems based on complex modal superposition method

In this paper, the response solutions of the damped fluid-conveying pipe system with elastic torsion constraints at both ends are analyzed. The pipe system considering gyroscopic effect induced by internal flow and damping effect is a typical damped gyroscopic system. This system cannot be decoupled in the modal space by the traditional modal analysis, and then the semi-analytical response solutions cannot be obtained by the classical modal superposition method. In order to remedy this problem, this paper proposes a new method based on complex modal superposition method and state-space method to give the semi-analytical response solutions of the damped fluid-conveying pipe system. The adjoint system is introduced through the concept of adjoint operator, and the orthogonality conditions of the damped fluid-conveying pipe system are derived in the state-space by using the eigenvalue relationship between the original system and the adjoint system. The Laplace transformation method is used to solve the eigenvalue problems of the original system and the adjoint system, and the implicit function equations about the eigenvalues and analytical eigenfunctions are obtained. According to the obtained orthogonality conditions, the semi-analytical response solutions of the system under arbitrary initial conditions and excitations are given, and the obtained solutions include transient and steady-state response solutions. In the numerical discussion part, the reliability and accuracy of the present method are verified. This research has positive significance for the dynamic analysis of fluid-conveying pipe systems, and the obtained orthogonality conditions have potential value for the nonlinear dynamics research.

Large Péclet number forced convection from a circular wire in a uniform stream: hybrid approximations at small Reynolds numbers

Abstract We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, $\mathrm{Re}\ll 1$ . This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of $\log \mathrm{Re}$ . We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math.55, 1484], based upon a robust asymptotic expansion in powers of $\mathrm{Re}$ . In that approximation, the “inner” streamfunction is provided by the product of a pre-factor $S$ , a slowly varying function of $\mathrm{Re}$ , with a $\mathrm{Re}$ -independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of $\log \mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse the transport problem in the limit of large Péclet number, $\mathrm{Pe}\gg 1$ . In that limit, transport is restricted to a narrow boundary layer about the cylinder surface – a province contained within the inner region of the flow problem. With $S$ appearing as a parameter, a similarity solution is readily constructed for the boundary-layer problem. It provides the Nusselt number as $0.5799(S\,\mathrm{Pe})^{1/3}$ . This asymptotic prediction is in remarkably close agreement with that of the numerical solution of the exact problem [Dennis, Hudson & Smith (1968) Phys. Fluids11, 933] even for moderate $\mathrm{Re}$ -values.

Stance and food activism on Arabic Twitter (X): a multimodal analysis

Abstract Expanding on classic theories of stancetaking and theorizing digital activism as nexus analysis, I examine how Omani Arab citizens used discourse and images to canalize a failed food boycott campaign on Twitter (X) into a united negative stance against alleged corruption. I highlight the role of multimodality, impoliteness, indirectness, intertextual references, participation framework and Arabic cultural practices in creating a multi-layered, accreted stance with manifold implicit functions (e.g., defining Omani identity, lamenting and signalling dissent). I argue for the need to widen the scope and level of analysis of stance acts to include interplays between texts, emoji, actions and images. I also highlight the role that stance can perform in expressing dissent and managing cultural face in the understudied Arabic context. I, therefore, demonstrate the multidimensionality of stance acts made visible by social media affordances and user creativity.

An algorithm for approximating implicit functions by polynomials without using higher order differentiability information

An algorithm for approximating implicit functions by polynomials without using higher order differentiability information

On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions

Abstract This study investigates the existence of solutions for implicit fractional differential equations with fractional-order integral boundary conditions. We create the required conditions to ensure unique solution and Ulam-Hyers-Rassias stability. We also give examples to highlight the major findings.

Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math. 49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.

Exploring Iterated Implicit Function Systems: Existence and Properties of Attractors

This paper investigates a type of iterated implicit function systems composed of equations [Formula: see text], where [Formula: see text] is a continuous function, and [Formula: see text] is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation [Formula: see text] containing the implicit function or being [Formula: see text]-contractive about [Formula: see text]. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

HQ3DAvatar: High-quality Implicit 3D Head Avatar

Multi-view volumetric rendering techniques have recently shown great potential in modeling and synthesizing high-quality head avatars. A common approach to capture full head dynamic performances is to track the underlying geometry using a mesh-based template or 3D cube-based graphics primitives. While these model-based approaches achieve promising results, they often fail to learn complex geometric details such as the mouth interior, hair, and topological changes over time. This article presents a novel approach to building highly photorealistic digital head avatars. Our method learns a canonical space via an implicit function parameterized by a neural network. It leverages multiresolution hash encoding in the learned feature space, allowing for high quality, faster training, and high-resolution rendering. At test time, our method is driven by a monocular RGB video. Here, an image encoder extracts face-specific features that also condition the learnable canonical space. This encourages deformation-dependent texture variations during training. We also propose a novel optical flow-based loss that ensures correspondences in the learned canonical space, thus encouraging artifact-free and temporally consistent renderings. We show results on challenging facial expressions and show free-viewpoint renderings at interactive real-time rates for a resolution of 480 x 270. Our method outperforms related approaches both visually and numerically. We will release our multiple-identity dataset to encourage further research.