arXiv subjects

Shenggao Zhou

Publications and source records attributed to Shenggao Zhou.

At least 19 recordsLinked to original sources

arXiv2025

Maximization of Supercapacitor Storage via Topology Optimization of Electrode Structures

As widely used electrochemical storage devices, supercapacitors deliver higher power density than batteries, but suffer from significantly lower energy density. In this work, we propose a topology optimization model for electrode structure to maximize energy storage in supercapacitors. The existence of minimizers to the resulting optimal control problem, which is constrained by a modified steady-state Poisson--Nernst--Planck system describing ionic electrodiffusion, has been theoretically established by using the direct method in the calculus of variation. Sensitivity analysis of the topology optimization model is performed to derive variational derivatives and corresponding adjoint equations. A gradient flow formulation discretized by a stabilized semi-implicit scheme is developed to solve the resulting topology optimization problem. Extensive numerical experiments present various porous electrode structures that own large area of electrode-electrolyte interface, demonstrating the effectiveness and robustness of the proposed topology optimization model and corresponding algorithm.

math.OC↗

arXiv2025

Mean-field Modeling of Social Interactions Using Classical Density Functional Theory

Incorporating social interactions is essential to an accurate modeling of epidemic spreading. This work proposes a novel local mean-field density functional theory model by using the sum-of-exponential approximation of convolution kernels for social interactions, which in turn converts the convolution terms into interaction potentials that are governed by the Debye-Hückel equation. Thanks to the local formulation of the proposed model, linear stability analysis is able to derive a novel instability condition associated with cross interactions. Global existence of the solution to the proposed model with a simplified self-repulsive interaction potential is established. Extensive numerical simulations are performed to assess the impact of cross social interactions on transmission and isolation, verify the instability conditions obtained from linear stability analysis, and provide theoretical guides for the control of disease spreading.

physics.soc-ph↗

arXiv2025

Intrinsic local Gauss's law preserving PIC method: A self-consistent field-particle update scheme for plasma simulations

In order to perform physically faithful particle-in-cell (PIC) simulations, the Gauss's law stands as a critical requirement, since its violation often leads to catastrophic errors in long-term plasma simulations. This work proposes a novel method that intrinsically enforces the Gauss's law for the Vlasov-Ampère/Vlasov-Poisson system without requiring auxiliary field corrections or specialized current deposition techniques. The electric field is managed to get updated locally and consistently with the motion of particles via splitting the motion into sub-steps along each dimension of the computational mesh. To further obtain a curl-free electric field, a local update scheme is developed to relax the electric-field free energy subject to the Gauss's law. The proposed method avoids solving the Poisson's or Ampère's equation, resulting in a local algorithm of linear complexity for each time step which can be flexibly combined with various temporal discretization for particle motion in PIC simulations. Theoretical analysis verifies that the proposed method indeed maintains the discrete Gauss's law exactly. Numerical tests on classical benchmarks, including the Landau damping, two-stream instability and Diocotron instability, demonstrate the key advantages of the proposed method. It is expected that the local nature of the proposed method makes it a promising tool in parallel simulations of large-scale plasmas.

physics.plasm-ph↗

arXiv2025

Shape Optimization of Supercapacitor Electrode to Maximize Charge Storage

We build a new mathematical model of shape optimization for maximizing ionic concentration governed by the multi-physical coupling steady-state Poisson-Nernst-Planck system. Shape sensitivity analysis is performed to obtain the Eulerian derivative of the cost functional. The Gummel fixed-point method with inverse harmonic averaging technique on exponential coefficient is used to solve efficiently the steady-state Poisson-Nernst-Planck system. Various numerical results using a shape gradient algorithm in 2d and 3d are presented.

math.OC↗

arXiv2024

Finite-Difference Approximations and Local Algorithm for the Poisson and Poisson-Boltzmann Electrostatics

We study finite-difference approximations of both Poisson and Poisson-Boltzmann (PB) electrostatic energy functionals for periodic structures constrained by Gauss' law and a class of local algorithms for minimizing the finite-difference discretization of such functionals. The variable of Poisson energy is the vector field of electric displacement and that for the PB energy consists of an electric displacement and ionic concentrations. The displacement is discretized at midpoints of edges of grid boxes while the concentrations are discretize at grid points. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy on each grid box, keeping Gauss' law satisfied. We prove that the energy functionals admit unique minimizers that are solutions to the corresponding Poisson's and charge-conserved PB equation, respectively. Local equilibrium conditions are identified to characterize the finite-difference minimizers of the discretized energy functionals. These conditions are the curl free for the Poisson case and the discrete Boltzmann distributions for the PB case, respectively. Next, we obtain the uniform bound with respect to the grid size h and O(h2)-error estimates in maximum norm for the finite-difference minimizers. The local algorithms are detailed, and a new local algorithm with shift is proposed to treat the general case of a variable coefficient for the Poisson energy. We prove the convergence of all these local algorithms, using the characterization of the finite-difference minimizers. Finally, we present numerical tests to demonstrate the results of our analysis.

math.NA↗

arXiv2024

Entropy Increasing Numerical Methods for Prediction of Reversible and Irreversible Heating in Supercapacitors

Accurate characterization of entropy plays a pivotal role in capturing reversible and irreversible heating in supercapacitors during charging/discharging cycles. However, numerical methods that can faithfully capture entropy variation in supercapacitors are still in lack. This work develops first-order and second-order finite-volume schemes for the prediction of non-isothermal electrokinetics in supercapacitors. Semi-implicit discretization that decouples temperature from ionic concentrations and electric potential results in an efficient first-order accurate scheme. Its numerical analysis theoretically establishes the unique solvability of the nonlinear scheme with the existence of positive ionic concentrations and temperature at discrete level. To obtain an entropy-increasing second-order scheme, a modified Crank-Nicolson approach is proposed for discretization of the logarithm of both temperature and ionic concentrations, which is employed to enforce numerical positivity. Moreover, numerical analysis rigorously demonstrates that both first-order and second-order schemes are able to unconditionally preserve ionic mass conservation and original entropy increase for a closed, thermally insulated supercapacitor. Extensive numerical simulations show that the proposed schemes have expected accuracy and robust performance in preserving the desired properties. Temperature oscillation in the charging/discharging processes is successfully predicted, unraveling a quadratic scaling law of temperature rising slope against voltage scanning rate. It is also demonstrated that the variation of ionic entropy contribution, which is the underlying mechanism responsible for reversible heating, is faithfully captured. Our work provides a promising tool in predicting reversible and irreversible heating in supercapacitors.

math.NA↗

arXiv2024

A second-order accurate, original energy dissipative numerical scheme for chemotaxis and its convergence analysis

This paper proposes a second-order accurate numerical scheme for the Patlak-Keller-Segel system with various mobilities for the description of chemotaxis. Formulated in a variational structure, the entropy part is novelly discretized by a modified Crank-Nicolson approach so that the solution to the proposed nonlinear scheme corresponds to a minimizer of a convex functional. A careful theoretical analysis reveals that the unique solvability and positivity-preserving property could be theoretically justified. More importantly, such a second order numerical scheme is able to preserve the dissipative property of the original energy functional, instead of a modified one. To the best of our knowledge, the proposed scheme is the first second-order accurate one in literature that could achieve both the numerical positivity and original energy dissipation. In addition, an optimal rate convergence estimate is provided for the proposed scheme, in which rough and refined error estimate techniques have to be included to accomplish such an analysis. Ample numerical results are presented to demonstrate robust performance of the proposed scheme in preserving positivity and original energy dissipation in blowup simulations.

math.NA↗

arXiv2023

Local Structure-Preserving Relaxation Method for Charged Systems on Unstructured Meshes

This work considers charged systems described by the modified Poisson--Nernst--Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the steady-state modified PNP equations poses numerical challenges due to the emergence of sharp boundary layers caused by small Debye lengths, particularly when local ionic concentrations reach saturation. To address this, we first reformulate the steady-state problem as a constraint optimization, where the ionic concentrations on unstructured Delaunay nodes are treated as fractional particles moving along edges between nodes. The electric fields are then updated to minimize the objective free energy while satisfying the discrete Gauss's law. We develop a local relaxation method on unstructured meshes that inherently respects the discrete Gauss's law, ensuring curl-free electric fields. Numerical analysis demonstrates that the optimal mass of the moving fractional particles guarantees the positivity of both ionic and solvent concentrations. Additionally, the free energy of the charged system consistently decreases during successive updates of ionic concentrations and electric fields. We conduct numerical tests to validate the expected numerical accuracy, positivity, free-energy dissipation, and robustness of our method in simulating charged systems with sharp boundary layers.

math.NA↗

arXiv2023

Second-order, Positive, and Unconditional Energy Dissipative Scheme for Modified Poisson-Nernst-Planck Equations

First-order energy dissipative schemes in time are available in literature for the Poisson-Nernst-Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

math.NA↗

arXiv2023

How Thermal Effect Regulates Cyclic Voltammetry of Supercapacitors

Cyclic voltammetry (CV) is a powerful technique for characterizing electrochemical properties of electrochemical devices. During charging-discharging cycles, thermal effect has profound impact on its performance, but existing theoretical models cannot clarify such intrinsic mechanism and often give poor prediction. Herein, we propose an interfacial model for the electro-thermal coupling, based on fundamentals in non-equilibrium statistical mechanics. By incorporating molecular interactions, our model shows a quantitative agreement with experimental measurements. The integral capacitance shows a first enhanced then decayed trend against the applied heat bath temperature. Such a relation is attributed to the competition between electrical attraction and Born repulsion via dielectric inhomogeneity, which is rarely understood in previous models. In addition, as evidenced in recent experimental CV tests, our model predicts the non-monotonic dependence of the capacitance on the bulk electrolyte density, further demonstrating its high accuracy. This work demonstrates a potential pathway towards next-generation thermal regulation of electrochemical devices.

cond-mat.mes-hall↗

arXiv2023

Asymptotic analysis on charging dynamics for stack-electrode model of supercapacitors

Supercapacitors are promising electrochemical energy storage devices due to their prominent performance in rapid charging/discharging rates, long cycle life, stability, etc. Experimental measurement and theoretical prediction on charging timescale for supercapacitors often have large difference. This work develops a matched asymptotic expansion method to derive the charging dynamics of supercapacitors with porous electrodes, in which the supercapacitors are described by the stack-electrode model. Coupling leading-order solutions between every two stacks by continuity of ionic concentration and fluxes leads to an ODE system, which is a generalized equivalent circuit model for zeta potentials, with the potential-dependent nonlinear capacitance and resistance determined by physical parameters of electrolytes, e.g., specific counterion valences for asymmetric electrolytes. Linearized stability analysis on the ODE system after projection is developed to theoretically characterize the charging timescale. The derived asymptotic solutions are numerically verified. Further numerical investigations on the biexponential charging timescales demonstrate that the proposed generalized equivalent circuit model, as well as companion linearized stability analysis, can faithfully capture the charging dynamics of symmetric/asymmetric electrolytes in supercapacitors with porous electrodes.

math.NA↗

arXiv2023

Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation

This work proposes a fast iterative method for local steric Poisson--Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

math.NA↗

arXiv2022

Structure-preserving numerical method for Maxwell-Ampère Nernst-Planck model

Charge dynamics play essential role in many practical applications such as semiconductors, electrochemical devices and transmembrane ion channels. A Maxwell-Ampère Nernst-Planck (MANP) model that describes charge dynamics via concentrations and the electric displacement is able to take effects beyond mean-field approximations into account. To obtain physically faithful numerical solutions, we develop a structure-preserving numerical method for the MANP model whose solution has several physical properties of importance. By the Slotboom transform with entropic-mean approximations, a positivity preserving scheme with Scharfetter-Gummel fluxes is derived for the generalized Nernst-Planck equations. To deal with the curl-free constraint, the dielectric displacement from the Maxwell-Ampère equation is further updated with a local relaxation algorithm of linear computational complexity. We prove that the proposed numerical method unconditionally preserves the mass conservation and the solution positivity at the discrete level, and satisfies the discrete energy dissipation law with a time-step restriction. Numerical experiments verify that our numerical method has expected accuracy and structure-preserving properties. Applications to ion transport with large convection, arising from boundary-layer electric field and Born solvation interactions, further demonstrate that the MANP formulation with the proposed numerical scheme has attractive performance and can effectively describe charge dynamics with large convection of high numerical cell Péclet numbers.

math.NA↗

arXiv2022

A Maxwell-Ampère Nernst-Planck Framework for Modeling Charge Dynamics

Understanding the properties of charge dynamics is crucial to many practical applications, such as electrochemical energy devices and transmembrane ion channels. This work proposes a Maxwell-Ampère Nernst-Planck (MANP) framework for the description of charge dynamics. The MANP model with a curl-free condition on the electric displacement is shown to be energy dissipative with respect to a convex free-energy functional, and demonstrated to be equivalent to the Poisson-Nernst-Planck model. By the energy dissipation law, the steady state of the MANP model reproduces the charge conserving Poisson--Boltzmann (PB) theory, providing an alternative energy stable approach to study the PB theory. In order to achieve the curl-free condition, a companion local curl-free relaxation algorithm, which is shown to naturally preserve the discrete Gauss's law and converge robustly with linear computational complexity, is developed for the MANP model. One of the main advantages of our development is that it can efficiently deal with space-dependent permittivity instead of solving the variable-coefficient Poisson's equation. Many-body effects such as ionic steric effects and Coulomb correlations can be incorporated within the MANP framework to derive modified MANP models for problems in which the mean-field approximation fails. Numerical results on the charge dynamics with such beyond mean-field effects in inhomogeneous dielectric environments are presented to demonstrate the performance of the MANP models in the description of charge dynamics, illustrating that the proposed MANP model provides a general framework for modeling charge dynamics.

physics.comp-ph↗

arXiv2022

Energetic Variational Approach for Prediction of Thermal Electrokinetics in Charging and Discharging Processes of Electrical Double Layer Capacitors

This work proposes a new variational, thermodynamically consistent model to predict thermal electrokinetics in electric double layer capacitors (EDLCs) by using an energetic variational approach. The least action principle and maximum dissipation principle from the non-equilibrium thermodynamics are employed to develop modified Nernst-Planck equations for non-isothermal ion transport with temperature inhomogeneity. Laws of thermodynamics are employed to derive a temperature evolution equation with heat sources due to thermal pressure and electrostatic interactions. Numerical simulations successfully predict temperature oscillation in the charging-discharging processes of EDLCs, indicating that the developed model is able to capture reversible and irreversible heat generations. The impact of ionic sizes and scan rate of surface potential on ion transport, heat generation, and charge current is systematically assessed in cyclic voltammetry simulations. It is found that the thermal electrokinetics in EDLCs cannot follow the surface potential with fast scan rates, showing delayed dynamics with hysteresis diagrams. Our work thus provides a useful tool for physics-based prediction of thermal electrokinetics in EDLCs.

physics.chem-ph↗

arXiv2022

A second order accurate numerical method for the Poisson-Nernst-Planck system in the energetic variational formulation

A second order accurate (in time) numerical scheme is proposed and analyzed for the Poisson-Nernst-Planck equation (PNP) system, reformulated as a non-constant mobility $H^{-1}$ gradient flow in the Energetic Variational Approach (EnVarA). The centered finite difference is taken as the spatial discretization. Meanwhile, the highly nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty to design a second order accurate scheme in time, while preserving the variational energetic structures. The mobility function is updated with a second order accurate extrapolation formula, for the sake of unique solvability. A modified Crank-Nicolson scheme is used to approximate the logarithmic term, so that its inner product with the discrete temporal derivative exactly gives the corresponding nonlinear energy difference; henceforth the energy stability is ensured for the logarithmic part. In addition, nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, with the help of the singularity associated with the logarithmic function. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNP system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are also presented.

math.NA↗

arXiv2022

Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson--Nernst--Planck Equations

The analysis of structure-preserving numerical methods for the Poisson--Nernst--Planck (PNP) system has attracted growing interests in recent years. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such results for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.

math.NA↗

arXiv2020

The Calculus of Boundary Variations and the Dielectric Boundary Force in the Poisson-Boltzmann Theory for Molecular Solvation

In a continuum model of the solvation of charged molecules in an aqueous solvent, the classical Poisson-Boltzmann (PB) theory is generalized to include the solute point charges and the dielectric boundary that separates the high-dielectric solvent from the low-dielectric solutes. With such a setting, we construct an effective electrostatic free-energy functional of ionic concentrations, where the solute point charges are regularized by a reaction field. We prove that such a functional admits a unique minimizer in a class of admissible ionic concentrations and that the corresponding electrostatic potential is the unique solution to the boundary-value problem of the dielectric-boundary PB equation. The negative first variation of this minimum free energy with respect to variations of the dielectric boundary defines the normal component of the dielectric boundary force. Together with the solute-solvent interfacial tension and van der Waals interaction forces, such boundary force drives an underlying charged molecular system to a stable equilibrium, as described by a variational implicit-solvent model. We develop an $L^2$-theory for the continuity and differentiability of solutions to elliptic interface problems with respect to boundary variations, and derive an explicit formula of the dielectric boundary force. With a continuum description, our result of the dielectric boundary force confirms a molecular-level prediction that the electrostatic force points from the high-dielectric and polarizable aqueous solvent to the charged molecules. Our method of analysis is general as it does not rely on any variational principles.

math.OC↗