Implicit Ordinary Differential Equations Research Articles

Bringing metasurfaces to analytical lens design: stigmatism and specific ray mapping.

We propose a method to design the exact phase profile of at least one metasurface in a stigmatic singlet that can be made to implement a desired ray mapping. Following the generalized vector law of refraction and Fermat's principle, we can obtain exact solutions for the required lens shape and phase profile of a phase gradient metasurface to respect particular ray conditions (e.g., Abbe sine) as if it were a freeform refractive element. To do so, the method requires solving an implicit ordinary differential equation. We present comparisons with Zemax simulations of illustrative designed lenses to confirm the anticipated optical behaviour.

Hyperbolic modeling of gradient damage and one-dimensional finite volume simulations

This study provides a new formulation of gradient damage model which allows an efficient explicit numerical solution of dynamics problems. The proposed methodology is based on an ”extended Lagrangian approach” developed by one of the authors for the nondissipative and dispersive shallow water equation. By using this strategy, the global minimization problem commonly derived for gradient damage models is recast as a purely local hyperbolic one with source terms that can be easily solved using finite volumes. The numerical solution of the governing system is then based on a fractional-step method consisting of a classical Godunov-type scheme and an implicit Ordinary Differential Equation solver for the local source terms. Numerical results are presented on one-dimensional multi-fragmentation and spalling tests for illustration purpose.

Bumble bee pollination and the wildflower/crop trade-off: When do wildflower enhancements improve crop yield?

Populations of wild insect pollinators such as bumble bees are threatened worldwide, which compromises pollinator-dependent crop yields. Intentionally planting wildflower patches in agricultural landscapes can support these populations and increase the pollination of nearby crops via the “spillover effect” (i.e., the exporter hypothesis), but may also distract bees from the crops and reduce their pollination via the “Circe principle” (i.e., the aggregation hypothesis). Considering the potentially high costs of these management strategies and the necessity to support wild insect pollinators in the Anthropocene, there is a pressing need to provide simulation tools that can inform best practices for wildflower plantings in agro-ecosystems. We developed a spatially implicit ordinary differential equations (ODEs) model specifically designed to determine the optimal wildflower-to-crop ratio as a function of wildflower patch (i) attractiveness, (ii) nutritional benefits, and (iii) blooming period relative to the crop. The model represents the population dynamics of a bumble bee colony and floral resources (crop and wildflower) in the landscape and nest during one harvesting season. We conduct a full factorial simulation experiment to identify the optimal characteristics of the wildflower patch (i.e., blooming period, attractiveness, relative abundance) that maximise crop yield via the enhancement of the number of bees pollinating crop flowers in a fictional blueberry farm. Our results suggest that providing highly attractive and nutritive wildflower resources before and not during the crop blooming season is the most beneficial strategy. When both flower types are in competition, pollination services can decrease, either when wildflowers are too attractive, or if they provide less benefits to the bees than the crop due to a trade-off between resources quality versus quantity.

Self-similar solutions for resistive diffusion, Ohmic heating, and Ettingshausen effects in plasmas of arbitrary β

Magneto-inertial fusion (MIF) approaches, such as the MagLIF experiment, use magnetic fields in dense plasma to suppress cross-field thermal conduction, attempting to reduce heat loss and trap alpha particles to achieve ignition. However, the magnetic field can introduce other transport effects, some of which are deleterious. An understanding of these processes is thus crucial for accurate modeling of MIF. We generalize past work exploiting self-similar solutions to describe transport processes in planar geometry and compare the model to the radiation-magnetohydrodynamics (MHDs) code Chimera. We solve the 1D extended MHD equations under pressure balance, making no assumptions about the ratio of magnetic and thermal pressures in the plasma. The resulting ordinary differential equation (ODE) boundary value problem is solved using a shooting method, combining an implicit ODE solver and a Newton–Raphson root finder. We show that the Nernst effect dominates over resistive diffusion in high β plasma, but its significance is reduced as the β decreases. On the other hand, we find that Ettingshausen and Ohmic heating effects are dominant in low β plasma and can be observable in even order unity β plasma, though in the presence of a strong temperature gradient heat conduction remains dominant. We then present a test problem for the Ohmic heating and Ettingshausen effects which will be useful to validate codes modeling these effects. We also observe that the Ettingshausen effect plays a role in preventing temperature separation when Ohmic heating is strong. Neglecting this term may lead to overestimates for the electron temperature at a vacuum–plasma interface, such as at the edge of a z-pinch. The model developed can be used to provide test problems with arbitrary boundary conditions for magnetohydrodynamics codes with the ability to freely switch on terms to compare their individual implementations.

New approach to the singular solution of implicit ordinary differential equations

<p style='text-indent:20px;'>In this paper, for a class of nonlinear implicit ordinary differential equation with variable coefficients, a new approach to get the analytical singular solutions are given both for first order and higher order equations. Sufficient conditions on singular solutions are strengthened into necessary and sufficient conditions in specific case. Finally, the new results are verified by practical examples and MATLAB illustrations compared with traditional methods.</p>

Antiperiodic boundary value problem for an implicit ordinary differential equation

The paper is devoted to the investigation of the antiperiodic boundary value problem for an implicit nonlinear ordinary differential equation $$f(t,x,\dot x)=0, \quad x(0)+x(\tau)=0.$$ We assume that the mapping $f:\mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^k$ defining the equation under consideration is smooth and satisfies the condition of uniform nondegeneracy of the first derivative $$ \inf \bigl\{ {\rm cov} f'_v (t,x,v):\, (t,x,v)\in \mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \bigr\}>0. $$ Here ${\rm cov} A$ is the Banach constant of the linear operator $A.$ The assumption of uniform non-degeneracy holds, in particular, for the mapping $f$ defining an explicit ordinary differential equation. For implicit equations, sufficient conditions for the existence of a solution to an antiperiodic boundary value problem are obtained, and estimates for solutions are found. Co\-ro\-l\-la\-ries for normal ordinary differential equations are formulated. To prove the main result, the original implicit equation is reduced to an explicit differential equation by applying a nonlocal implicit function theorem. Then we prove an auxiliary assertion on the solvability of the equation $x+\psi(x)=0,$ which is an analog of Brouwer's fixed point theorem. It is shown that the mapping $\psi,$ that assigns the value of the solution of the Cauchy problem at the point $\tau$ to an arbitrary initial point $x_0,$ is well defined and satisfies the assumptions of the auxiliary statement. This reasoning completes the proof of the existence of a solution to the boundary value problem.

A Robust and Sleek Electrochemical Battery Model Implementation: A MATLAB® Framework

A Doyle-Fuller-Newman electrochemical battery model implementation in a robust and sleek MATLAB® framework for lithium-ion batteries as an open-access MATLAB code is presented. The Doyle-Fuller-Newman (DFN) model, in the form of partial differential equations, is first numerically discretized then converted to a differential algebraic equation (DAE). The most efficient way to implement a DAE system is through the adoption of standard DAE solvers provided by available commercial software. MATLAB® is a widely used software in the control community, and to the best of our knowledge, its standard solvers have failed to successfully simulate the DFN model when the battery undergoes high C-rates of operations. One critical issue with DFN model simulation is related to the inconsistency of initial conditions. In 2015, a robust single-step iteration-free initialization approach, enabling solving DAE systems using a standard solver using Maple® symbolic environment, was proposed by Lawder et al. A symbolic environment enables direct and efficient derivation of implicit ordinary differential equations from algebraic equations during initialization. We perform the single-step iteration-free initialization approach in MATLAB® environment by adopting the MATLAB® symbolic toolbox and simulate the DFN model with the ode15s solver. This framework allows users to robustly simulate the DFN model and identify model parameters directly after numerical discretization, utilizing a standard solver.

Fully implicit spectral boundary integral computation of red blood cell flow

This paper is on an implicit time integration scheme for simulation of red blood cell (RBC) flow in an ambient fluid. The intra- and extracellular plasmas are modeled as Stokes flows and represented by boundary integral equations (BIE) written in a weakly singular form. The cell membrane is modeled as a thin elastic shell. Expressed in this way, the RBC flow model constitutes an implicit ordinary differential equation (IODE) in the cell shape. The cell shape and velocity field are discretized spatially by a spectral approach using spherical harmonic basis functions. It is then convenient to express the BIE in the Galerkin form with the spherical harmonics themselves as test functions. The key aspect in this paper is the recognition of the IODE structure of the RBC flow model and consequent application of a multi-step implicit solver for time integration. As with any implicit solver, a nonlinear equation in the cell shape is solved at each time step, for which Newton's method is applied. This requires the Jacobian of the IODE, or equivalently computation of Jacobian-vector products. An important contribution is the formulation of such Jacobian-vector products as evaluating a second BIE. The original weakly singular form is crucial in facilitating this formulation. The implicit solver employs variable order and adaptive time stepping controlled by truncation error and convergence of Newton iterations. Numerical examples show that larger time steps are possible and that the number of matrix-vector products is comparable to explicit methods. Source code is provided in the online supplementary material.

A grid-free approach for simulating sweep and cyclic voltammetry.

We present a computational approach to simulate linear sweep and cyclic voltammetry experiments that does not require a discretized grid in space to quantify diffusion. By using a Green's function solution coupled to a standard implicit ordinary differential equation solver, we are able to simulate current and redox species concentrations using only a small grid in time. As a result, where benchmarking is possible, we find that the current method is faster than (and quantitatively identical to) established techniques. The present algorithm should help open the door for studying adsorption effects in inner sphere electrochemistry.

Compact sparse symbolic Jacobian computation in large systems of ODEs

This work introduces a novel algorithm that automatically produces computer code for the calculation of sparse symbolical Jacobian matrices. More precisely, given the code for computing a function f depending on a set of state (independent) variables x, where the code makes use of intermediate algebraic (auxiliary) variables a(x), the algorithm automatically produces the code for the symbolic computation of the matrix J=∂f/∂x in sparse representation.A remarkable feature of the algorithm developed is that it can deal with iterative definitions of the functions preserving the iterative representation during the whole process up to the final Jacobian computation code. That way, in presence of arrays of functions and variables, the computational cost of the code generation and the length of the generated code does not depend on the size of those arrays. This feature is achieved making use of Set–Based Graph representation.The main application of the algorithm is the simulation of large scale dynamical systems with implicit Ordinary Differential Equation (ODE) solvers like CVODE-BDF, whose performance are greatly improved when they are invoked using a sparse Jacobian matrix. However, the algorithm can be used in a more general context for solving large systems of nonlinear equations.The paper, besides introducing the algorithm, discusses some aspects of its implementation in a general purpose ODE solver front-end and analyzes some results obtained.

A Wave Digital Newton-Raphson Method for Virtual Analog Modeling of Audio Circuits with Multiple One-Port Nonlinearities

The digital implementation of a nonlinear audio circuit often employs the Newton-Raphson (NR) method for solving the corresponding system of implicit ordinary differential equations in the discrete-time domain. Although its quadratic convergence speed makes NR attractive for real-time audio applications, quadratic convergence is not always guaranteed, since it depends on initial conditions, and also divergence might occur. For this reason, especially in the context of Virtual Analog modeling, techniques for increasing the robustness of NR are in order. Among the various approaches, the Wave Digital (WD) formalism recently showed potential to rethink traditional circuit simulation methods. In this manuscript, we discuss an original formulation of the NR method in the WD domain for the solution of audio circuits with multiple one-port nonlinearities. We provide an in-depth theoretical analysis of the proposed iterative method and we show how its quadratic convergence strongly depends on the free parameters (called port resistances) introduced when modeling the reference circuit in the WD domain. In particular, we demonstrate that the size of the basin where the WD NR solver can be initialized to converge on a solution with quadratic speed is a function of the free parameters. We also show that by setting each port resistance value as close as possible to the derivative w.r.t. current of the nonlinear element v-i characteristic we keep the basin size large. We finally implement an audio ring modulator circuit with four diodes in order to test the proposed iterative method.

Dynamic linepack depletion models for natural gas pipeline networks

Given the critical role played by natural gas in providing electricity, heat, and other essential services, better models are needed to understand the dynamics of natural gas networks during extreme events. This paper aims at establishing appropriate and fast simulation models to capture the slow dynamics of linepack depletion for ideal isothermal natural gas pipeline networks. Instead of solving partial differential equations (PDE) on a large scale, three alternative implicit ordinary differential equation (ODE) simulation techniques are derived and discussed. The first one is commonly used in the literature with a slack node assumption. We show that the system of equations associated with this model is degenerate when flux injections are controlled (i.e. specified) at all nodes. To recover regularity under such a condition, two novel implicit ODE models are proposed with different techniques on boundary conditions. They are easy to derive and efficient to simulate by standard ODE solvers. More importantly, they present useful frameworks for analyzing how networks respond to system-wide mass flux imbalances. These techniques offer different alternatives for simulating system dynamics based on how sources and loads are chosen to be modeled, and they are all proven to be regular (non-degenerate) in tree-structured networks. These proposed techniques are all tested on the 20-node Belgium network. The simulation results show that the conventional model with the slack node assumption cannot effectively capture linepack depletion under long term system-wide mass flux imbalance, while the proposed models can characterize the network behavior until the linepack is completely depleted.

SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics.

Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov–Zhabotinsky (BZ) reaction.

A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce.

Computationally Efficient Nonlinear One- and Two-Track Models for Multitrailer Road Vehicles

This paper presents nonlinear mathematical models of one- and two-track multitrailer vehicles. We derive nonlinear equations of motion in the form of a system of implicit ordinary differential equations (ODEs) by using Lagrangian mechanics. The system of ODEs has the minimum number of states and equations that enables efficient computations yet maintains the most important nonlinear vehicle dynamic behavior and allows actuator coordination and energy consumption evaluation. As examples, we build different models of a 4-unit long combination vehicle, i.e., two-track 11-axle and single-track 6-axle nonlinear models as well as a linear single-track 6-axle model. We compare the performance of these models to experimental data of different driving maneuvers. The nonlinear single-track model demonstrates close dynamic behavior to the experiment, which makes it an efficient alternative to the two-track model. The vehicle equations can be generated automatically by using the code provided in this paper and subsequently used for conducting frequency analysis, evaluating energy consumption, deriving performance measures from simulations, and facilitating optimal control applications that involve combined steering, braking and propulsion control.

On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.

Simulation and optimization of dynamic flux balance analysis models using an interior point method reformulation

This work presents a novel, differentiable, way of solving dynamic Flux Balance Analysis (dFBA) problems by embedding flux balance analysis of metabolic network models within lumped bulk kinetics for biochemical processes. The proposed methodology utilizes transformation of the bounds of the embedded linear programming problem of flux balance analysis via a logarithmic barrier (interior point) approach. By exploiting the first-order optimality conditions of the interior-point problem, and with further transformations, the approach results in a system of implicit ordinary differential equations. Results from four case studies, show that the CPU and wall-times obtained using the proposed method are competitive with existing state-of-the art approaches for solving dFBA simulations, for problem sizes up to genome-scale. The differentiability of the proposed approach allows, using existing commercial packages, its application to the optimal control of dFBA problems at a genome-scale size, thus outperforming existing formulations as shown by two dynamic optimization case studies.

Formulation and Implementation of Nonlinear Integral Equations to Model Neural Dynamics Within the Vertebrate Retina

Existing computational models of the retina often compromise between the biophysical accuracy and a hardware-adaptable methodology of implementation. When compared to the current modes of vision restoration, algorithmic models often contain a greater correlation between stimuli and the affected neural network, but lack physical hardware practicality. Thus, if the present processing methods are adapted to complement very-large-scale circuit design techniques, it is anticipated that it will engender a more feasible approach to the physical construction of the artificial retina. The computational model presented in this research serves to provide a fast and accurate predictive model of the retina, a deeper understanding of neural responses to visual stimulation, and an architecture that can realistically be transformed into a hardware device. Traditionally, implicit (or semi-implicit) ordinary differential equations (OES) have been used for optimal speed and accuracy. We present a novel approach that requires the effective integration of different dynamical time scales within a unified framework of neural responses, where the rod, cone, amacrine, bipolar, and ganglion cells correspond to the implemented pathways. Furthermore, we show that adopting numerical integration can both accelerate retinal pathway simulations by more than 50% when compared with traditional ODE solvers in some cases, and prove to be a more realizable solution for the hardware implementation of predictive retinal models.

Two methods of data reconciliation for pipeline networks

The paper compares the two most common methods of data reconciliation for pipeline networks. The first method, an unscented Kalman filter (UKF), uses a system of nonlinear implicit ordinary differential equations derived from the governing partial differential equations. The second method, a quadratic program, relies on a transformation of the system of nonlinear ordinary differential equations into a set of linear difference equations, with the linearization optimized for known pressure and flow ranges using a novel linearization technique. Both the UKF and the quadratic programming approaches for data reconciliation in gas pipeline networks are viable for networks of small to moderate size. Given the reduced number of simplifying assumptions and the resulting improved accuracy, the UKF may be preferable when the computational problem is tractable. The quadratic programming approach is faster, accepts lower fidelity models, and provides acceptable accuracy.

Accurate modeling and analysis of a bio-inspired isolation system: with application to on-orbit capture

Abstract The accurate dynamical model of a bio-inspired isolation (BII) system with complete consideration of the kinetic and potential energies of all components is established for the first time. Owing to the inclusion of energies pertaining to the rods and joints in the modeling process, the equations of motion of the accurate BII model are virtually governed by a set of implicit ordinary differential equations (IODEs), which is totally different from the simplified model whose governing equations are standard ODEs. Since the accurate model cannot be transformed into the explicit form, viz. x = f ( x , t ) , it cannot be handled via the traditional numerical integration methods, e.g. the Runge-Kutta method. In order to conquer this difficulty, the harmonic balance method and the Matlab’s ODE15i code are employed to solve the periodic and aperiodic solutions, respectively. More importantly, the comparisons of dynamical responses as well as isolation performance between the accurate and simplified BII models are conducted in two situations. For the situation when system exhibits simple responses, the accurate and the simplified models generate qualitatively the same pattern of results. The only discrepancy is that the latter model produces an overestimated vibration. For the complex responses, an essential distinction occurs between the two models, indicating that using the accurate BII model is extremely important. Furthermore, the efficiency of the BII isolator is verified through comparing with the traditional spring-mass-damper (SMD) system. The influences of system parameters, such as damping, layer number, assembly angle, spring stiffness, etc., on the isolation performance have been investigated. Finally, numerical examples verify the efficiency of the present BII isolator in both ground and space environments.