arXiv subjects

B. Shanker

Publications and source records attributed to B. Shanker.

At least 19 recordsLinked to original sources

arXiv2025

Sparse Operator-Adapted Wavelet Decomposition Using Polygonal Elements for Multiscale FEM Problems

We develop a sparse multiscale operator-adapted wavelet decomposition-based finite element method (FEM) on unstructured polygonal mesh hierarchies obtained via a coarsening procedure. Our approach decouples different resolution levels, allowing each scale to be solved independently and added to the entire solution without the need to recompute coarser levels. At the finest level, the meshes consist of triangular elements which are geometrically coarsened at each step to form convex polygonal elements. Smooth field regions of the domain are solved with fewer, larger, polygonal elements, whereas high-gradient regions are represented by smaller elements, thereby improving memory efficiency through adaptivity. The proposed algorithm computes solutions via sequences of hierarchical sparse linear-algebra operations with nearly linear computational complexity.

physics.comp-ph↗

arXiv2025

Hierarchical Finite-Element Analysis of Multiscale Electromagnetic Problems via Sparse Operator-Adapted Wavelet Decomposition

In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches, while capable of achieving the desired accuracy without requiring a complete re-meshing of the computational domain, inherently couple different resolution levels. This coupling requires recomputation of coarser-level solutions whenever finer details are added to improve accuracy, resulting in substantial computational overhead. Our proposed method addresses this issue by decoupling resolution levels. This feature enables independent computations at each scale that can be incorporated into the solutions to improve accuracy whenever needed, without requiring re-computation of coarser-level solutions. The main algorithm is hierarchical, constructing solutions from finest to coarser levels through a series of sparse matrix-vector multiplications. Due to its sparse nature, the overall computational complexity of the algorithm is nearly linear. Moreover, Krylov subspace iterative solvers are employed to solve the final linear equations, with ILU preconditioners that enhance solver convergence and maintain overall computational efficiency. The numerical experiments presented in this article verify the high precision and nearly linear computational complexity of the proposed algorithm.

physics.comp-ph↗

arXiv2023

A Transient Port-Extraction Technique for Antenna Feed Optimization

Optimization of strongly non-linear tightly coupled feeds attached to antennas is a challenging problem from a purely computational perspective. One can imagine that an optimization would (a) need to be in the time domain, and (b) has to be self-consistently coupled with the linear antenna (or electromagnetic) system. These two imply that the cost of optimization is governed by the need to repeatedly evaluate the fully coupled cost function. This paper leverages a recently developed transient port-extraction technique to circumvent this challenge and is agnostic to the optimization scheme. This approach provides a representation of the entire linear electromagnetic system at the port and can readily integrate with any non-linear circuit analysis and optimization methodology. In this paper, we demonstrate optimization of linear and non-linear circuit feed parameters that are tightly coupled to broadband radiating systems.

math.OC↗

arXiv2023

A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations

The state of art of charge-conserving electromagnetic finite element particle-in-cell has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog based solvers (and their other incarnations) are only conditionally stable. Recent advances have made Electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) methods built around unconditionally stable time stepping schemes were shown to conserve charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss' Laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side-effect as earlier--they are conditionally stable. Here we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor-corrector method for Newton's equations. As we will show via numerical experiments, the proposed method conserves energy within a PIC scheme, has an unconditionally stable EM solve, solves Newton's equations to much higher accuracy than a traditional Boris solver and conserves charge to machine precision. We further demonstrate benefits compared to other polynomial methods to solve Newton's equations, like the well known Boris push.

math.NA↗

arXiv2022

Analytic preconditioners for decoupled potential integral equations and wideband analysis of scattering from PEC objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calder\'{o}n-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

physics.comp-ph↗

arXiv2022

An Envelope Tracking Approach for Particle in Cell Simulations

The state of the art in electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) has advanced significantly in the last few years. These have included understanding function spaces that must be used to represent sources and fields consistently, and how currents should be evolved in space and time. In concert, these achieve satisfaction of Gauss laws. All of these, were restricted to conditionally stable explicit time stepping. More recently, there has been advances to the state of art. It is now possible to use a implicit EM FEMPIC method while satisfying Gauss laws to machine precision. This enables choosing time step sizes dictated by physics as opposed to geometry. In this paper, we take this a step further. For devices characterized by a narrowband high frequency response, choosing a time-step size based on the highest frequency of interest is considerably expensive. In this paper, we use methods derived from envelope tracking to construct an EM-FEMPIC method that analytically provides for the high-frequency oscillations of the system, allowing for analysis at considerable coarser time-step sizes even in the presence of non-linear effects from active media such as plasmas. Consequentially, we demonstrate how the pointwise metric used for measuring satisfaction of Gauss Laws breaks down when prescribing analytical fast fields and provide a thorough analysis of how charge conservation can be measured. Through a number of examples, we demonstrate that the proposed approach retains the accuracy the regular scheme while requiring far fewer time steps.

physics.comp-ph↗

arXiv2022

Transient dynamics of subradiance and superradiance in open optical ensembles

We introduce a computational Maxwell-Bloch framework for investigating out-of-equilibrium optical emitters in open systems. To do so, we compute the pulse-induced dynamics of each emitter from fundamental light-matter interactions and self-consistently calculate their radiative coupling, including phase inhomogeneity from propagation effects. This semiclassical framework is applied to open quantum dots systems with different densities and dipolar coupling. We observe signatures of superradiant behavior, such as directionality and faster decay, as well as subradiant emission. We compare and discuss the computed light emission obtained with our method and a Master equation approach. Our framework enables quantitative investigations of large optical ensembles in the time domain and could be used to design new systems with enhanced superradiant and subradiant properties.

physics.optics↗

arXiv2022

Domain Decomposition Framework for Maxwell Finite Element Solvers and Application to PIC

The most popular methods for self-consistent simulation of fields interacting with charged species is using finite difference time domain (FDTD) methods together with Newton's laws of motion to evolve locations and velocities of particles. Despite their popularity, the limitation of FDTD particle in cell (EM-FDTDPIC) methods are well known. To address these, there has been significant interest over the past decade in exploring alternatives. In the past few years, the advances in electromagnetic finite element methods for particle in cell (EM-FEMPIC) has advanced by leaps and bounds. The mathematics necessary for implicit FEM methods that are unconditionally stable and charge conserving are now well understood. Some of these advances are more recent. The next bottleneck necessary to make EM-FEMPIC competitive with FDTD based scheme is overcoming computational cost. Our approach to resolving this challenge is develop two different finite element tearing and integration approaches, and using these to create domain decomposition schemes for EM-FEMPIC. Details of the proposed methodology are presented as well as a number of results that demonstrates charge conservation as well as amelioration of costs for a number of problems.

physics.comp-ph↗

arXiv2022

A Complete Helmholtz Decomposition on Multiply Connected Subdivision Surfaces and Its Application to Integral Equations

The analysis of electromagnetic scattering in the isogeometric analysis (IGA) framework based on Loop subdivision has long been restricted to simply-connected geometries. The inability to analyze multiply-connected objects is a glaring omission. In this paper, we address this challenge. IGA provides seamless integration between the geometry and analysis by using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply-connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free and curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply-connected structures. In this paper, we effect this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these basis recovers curl-free, divergence-free, and curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.

math.NA↗

arXiv2021

Higher Order Charge Conserving Electromagnetic Finite Element Particle in Cell Method

Until recently, electromagnetic finite element PIC (EM-FEMPIC) methods that demonstrated charge conservation used explicit field solvers. It is only recently, that a series of papers developed the mathematics necessary for charge conservation within an implicit field solve and demonstrated for a number of examples. This permits using time steps sizes that are necessary to capture the physics as opposed to being restricted to those constrained by geometry. One aspect that is missing is higher order basis functions to represent both fields and particles. Higher order basis can be particularly helpful in effectively capturing complex field layouts with fewer degrees of freedom. Developing a framework for higher order EM-FEMPIC that maintains stability, improves accuracy, and conserves charge is the principal goal of this paper. A number of results are presented that attest to its efficacy.

physics.comp-ph↗

arXiv2021

Iso-geometric Integral Equation Solvers and their Compression via Manifold Harmonics

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are continuous, but their normals are discontinuous at the boundary between triangular tessellations of control nodes, or patches, with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design are second order differentiable almost everywhere on the surfaces. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.

math.NA↗

arXiv2021

Port Parameter Extraction Based Self Consistent Coupled EM-Circuit FEM Solvers

Self consistent solution to electromagnetic (EM)-circuit systems is of significant interest for a number of applications. This has resulted in exhaustive research on means to couple them. In time domain, this typically involves a tight integration (or coupling) with field and non-linear circuit solvers. This is in stark contrast to coupled analysis of linear/weakly non-linear circuits and EM systems in frequency domain. Here, one typically extracts equivalent port parameters that are then fed into the circuit solver. Such an approach has several advantages; (a) the number of ports is typically smaller than the number of degrees of freedom, resulting in cost savings; (b) is circuit agnostic; (c) can be integrated with a variety of device models. Port extraction is tantamount to obtaining impulse response of the linear EM system. In time domain, the deconvolution required to effect this is unstable. Recently, a novel approach was developed for time domain integral equations to overcome this bottleneck. We extend this approach to time domain finite element method, and demonstrate its utility via a number of examples; significantly, we demonstrate that self consistent solutions obtained using either a fully coupled or port extraction is identical to the desired precision for non-linear circuit systems. This is shown within a nodal network. We also demonstrate integration of port extracted data directly with drift diffusion equation to model device physics.

eess.SP↗

arXiv2021

Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics, provide the means to smoothly interpolate data on a manifold. They are highly effective, specifically as it relates to geometry representation and editing; manifold harmonics form a natural basis for multi-resolution representation (and editing) of complex surfaces and functioned defined therein. In this paper, we seek to develop the framework to exploit the benefits of manifold harmonics for shape reconstruction. To this end, we develop a highly compressible, multi-resolution shape reconstruction scheme using manifold harmonics. The method relies on subdivision basis sets to construct both boundary element isogeometric methods for analysis and surface finite elements to construct manifold harmonics. We pair this technique with the volumetric source reconstruction method to determine an initial starting point. Examples presented highlight efficacy of the approach in the presence of noisy data, including significant reduction in the number of degrees of freedom for complex objects, the accuracy of reconstruction, and multi-resolution capabilities.

math.NA↗

arXiv2021

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

Development of particle in cell methods using finite element based methods (FEMs) have been a topic of renewed interest; this has largely been driven by (a) the ability of finite element methods to better model geometry, (b) better understanding of function spaces that are necessary to represent all Maxwell quantities, and (c) more recently, the fundamental rubrics that should be obeyed in space and time so as to satisfy Gauss' laws and the equation of continuity. In that vein, methods have been developed recently that satisfy these equations and are agnostic to time stepping methods. While is development is indeed a significant advance, it should be noted that implicit FEM transient solvers support an underlying null space that corresponds to a gradient of a scalar potential $\nabla \Phi(\textbf{r})$ (or $t \nabla \Phi (\textbf{r})$ in the case of wave equation solvers). While explicit schemes do not suffer from this drawback, they are only conditionally stable, time step sizes are mesh dependent, and very small. A way to overcome this bottleneck, and indeed, satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation on a tessellation. In the re-formulation presented, we strictly satisfy the equation of continuity and Gauss' laws for both the electric and magnetic flux densities. Results demonstrating the efficacy of this scheme will be presented.

physics.comp-ph↗

arXiv2021

Time Integrator Agnostic Charge Conserving Finite Element PIC

Developing particle-in-cell (PIC) methods using finite element basis sets, and without auxiliary divergence cleaning methods, was a long standing problem until recently. It was shown that if consistent spatial basis functions are used, one can indeed create a methodology that was charge conserving, albeit using a leap-frog time stepping method. While this is a significant advance, leap frog schemes are only conditionally stable and time step sizes are closely tied to the underlying mesh. Ideally, to take full advantage of advances in finite element methods (FEMs), one needs a charge conserving PIC methodology that is agnostic to the time stepping method. This is the principal contribution of this paper. In what follows, we shall develop this methodology, prove that both charge and Gauss' laws are discretely satisfied at every time step, provide the necessary details to implement this methodology for both the wave equation FEM and Maxwell Solver FEM, and finally demonstrate its efficacy on a suite of test problems. The method will be demonstrated by single particle evolution, non-neutral beams with space-charge, and adiabatic expansion of a neutral plasma, where the debye length has been resolved, and real mass ratios are used.

physics.plasm-ph↗

arXiv2021

Rubrics for Charge Conserving Current Mapping in Finite Element Particle in Cell Methods

Modeling of kinetic plasmas using electromagnetic particle in cell methods (EM-PIC) is a problem that is well worn, in that methods developed have been used extensively both understanding physics and exploiting them for device design. EM-PIC tools have largely relied on finite difference methods coupled with particle representations of the distribution function. Refinements to ensure consistency and charge conservation have largely been an ad-hoc efforts specific to finite difference methods. Meanwhile, solution methods for field solver have grown by leaps and bounds with significant performance metrics compared to finite difference methods. Developing new EM-PIC computational schemes that leverage modern field solver technology means re-examining analysis framework necessary for self-consistent EM-PIC solution. In this paper, we prescribe general rubrics for charge conservation, demonstrate how these are satisfied in conventional finite difference PIC as well as finite element PIC, and prescribe a novel charge conserving finite element PIC. Our effort leverages proper mappings on to de-Rham sequences and lays a groundwork for understanding conditions that must be satisfied for consistency. Several numerical results demonstrate the applicability of these rubrics.

physics.comp-ph↗

arXiv2021

A Set of Benchmark Tests for Validation of 3D Particle In Cell Methods

While the particle-in-cell (PIC) method is quite mature, verification and validation of both newly developed methods and individual codes has largely focused on an idiosyncratic choice of a few test cases. Many of these test cases involve either one- or two-dimensional simulations. This is either due to availability of (quasi) analytic solutions or historical reasons. Additionally, tests often focus on investigation of particular physics problems, such as particle emission or collisions, and do not necessarily study the combined impact of the suite of algorithms necessary for a full featured PIC code. As three dimensional (3D) codes become the norm, there is a lack of benchmarks test that can establish the validity of these codes; existing papers either do not delve into the details of the numerical experiment or provide other measurable numeric metrics (such as noise) that are outcomes of the simulation. This paper seeks to provide several test cases that can be used for validation and bench-marking of particle in cell codes in 3D. We focus on examples that are collisionless, and can be run with a reasonable amount of computational power. Four test cases are presented in significant detail; these include, basic particle motion, beam expansion, adiabatic expansion of plasma, and two stream instability. All presented cases are compared either against existing analytical data or other codes. We anticipate that these cases should help fill the void of bench-marking and validation problems and help the development of new particle in cell codes.

physics.comp-ph↗

arXiv2019

Acceleration techniques for semiclassical Maxwell-Bloch systems: An application to discrete quantum dot ensembles

The solution to Maxwell-Bloch systems using an integral-equation-based framework has proven effective at capturing collective features of laser-driven and radiation-coupled quantum dots, such as light localization and modifications of Rabi oscillations. Importantly, it enables observation of the dynamics of each quantum dot in large ensembles in a rigorous, error-controlled, and self-consistent way without resorting to spatial averaging. Indeed, this approach has demonstrated convergence in ensembles containing up to $10^4$ interacting quantum dots. Scaling beyond $10^4$ quantum dots tests the limit of computational horsepower, however, due to the $\mathcal{O}(N_t N_s^2)$ scaling (where $N_t$ and $N_s$ denote the number of temporal and spatial degrees of freedom). In this work, we present an algorithm that reduces the cost of analysis to $\mathcal{O}(N_t N_s \log^2 N_s)$. While the foundations of this approach rely on well-known particle-particle/particle-mesh and adaptive integral methods, we add refinements specific to transient systems and systems with multiple spatial and temporal derivatives. Accordingly, we offer numerical results that validate the accuracy, effectiveness and utility of this approach in analyzing the dynamics of large ensembles of quantum dots.

physics.comp-ph↗