arXiv subjects

Biswajit Khara

Publications and source records attributed to Biswajit Khara.

11 recordsLinked to original sources

arXiv2026

A Semi-Implicit Variational Multiscale Formulation for the Incompressible Navier-Stokes Equations via Exact Adjoint Linearization

A semi-implicit, residual-based variational multiscale (VMS) formulation is developed for the incompressible Navier-Stokes equations. The convection term is linearized using an extrapolated (Oseen-type) convecting velocity, producing a linear advection operator whose adjoint can be written exactly. Because of this exact adjoint, unresolved-scale contributions enter the weak form without spatial derivatives of the fine-scale velocity, thereby eliminating the case-by-case adjustments that often accompany nonlinear residual-based VMS implementations. The formulation is presented for a generalized linear convection operator encompassing the convective, skew-symmetric, and divergence forms. Since the discrete method is linear by construction and monolithic for velocity and pressure, each time step requires only one linear solve, reducing wall-clock time by a factor of $2$ to $5$ relative to fully implicit nonlinear formulations while maintaining comparable accuracy. Temporal convergence is verified, and validation is performed on the lid-driven cavity, flow past a cylinder, turbulent channel flow, and flow over a NACA0012 airfoil at a high Reynolds number, demonstrating the efficiency of the proposed approach on problems of practical scale.

physics.flu-dyn↗

arXiv2026

A Helmholtz-Leray projection method with variational multiscale stabilization for the Navier-Stokes equations

The Galerkin finite element formulation of the incompressible Navier-Stokes equations presents two principal challenges: maintaining stable velocity-pressure coupling and controlling instability in advection-dominated regimes. Moreover, the monolithic formulation produces a coupled nonlinear saddle-point problem. In this work, we present a residual-based variational multiscale (VMS) stabilization of an incremental Helmholtz-Leray projection method that replaces this coupled saddle-point problem with a nonlinear velocity predictor, a pressure Poisson equation, and a velocity projection. The multiscale decomposition is applied only to the predicted velocity; neither the pressure nor the corrected, weakly divergence-free velocity is decomposed into coarse and fine scales. The modeled velocity fine scale contributes consistently to all three subproblems, introducing SUPG-like stabilization in the momentum predictor and a PSPG-like residual contribution in the pressure Poisson equation. We provide a formal error decomposition that separates the BDF2 time-discretization, projection-splitting, and spatial-VMS errors and, under stated stability and spatial-approximation assumptions, yields a combined velocity error estimate with second-order temporal accuracy. Numerical results for manufactured solutions, lid-driven cavity flow, flow past a cylinder, and the Taylor-Green vortex agree closely with established reference data. Comparisons with monolithic VMS indicate that omitting the pressure fine scale reduces drag overprediction and excess modeled dissipation, at the cost of increased divergence error. In the Taylor-Green tests, the projection formulation also reduces the average solution time per step by factors ranging from approximately $1.3\times$ to $2.7\times$ under identical solver settings.

math.NA↗

arXiv2026

Proximal Galerkin for Phase Field Fracture

The phase-field method has emerged as a powerful tool for simulating fracture mechanics, yet it presents significant numerical challenges, particularly regarding the enforcement of physical constraints such as irreversibility and boundedness of the phase-field variable. This work proposes the proximal Galerkin (PG) methodology as a robust and efficient framework for solving phase-field fracture problems. By reformulating the inequality-constrained optimization problem into a sequence of saddle-point problems involving latent variables, the PG method rigorously enforces the physical bounds of the phase-field variable and naturally handles the irreversibility condition. This approach is directly applicable to both static and dynamic phase-field fracture problems. The numerical results demonstrate that the PG framework accurately reproduces theoretical predictions and experimental observations, while offering a unified, mathematically consistent treatment of the constraints inherent to phase-field fracture modeling.

math.NA↗

arXiv2024

Space-time finite element analysis of the advection-diffusion equation using Galerkin/least-square stabilization

We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete variational problem. In the full space-time formulation, time is considered another dimension, and the time derivative is interpreted as an additional advection term of the field variable. We derive a priori error estimates and illustrate spatio-temporal convergence with several numerical examples. We also derive a posteriori error estimates, which coupled with adaptive space-time mesh refinement provide efficient and accurate solutions. The accuracy of the space-time solutions is illustrated against analytical solutions as well as against numerical solutions using a conventional time-marching algorithm.

math.NA↗

arXiv2024

Solving fluid flow problems in space-time with multiscale stabilization: formulation and examples

We present a space-time continuous-Galerkin finite element method for solving incompressible Navier-Stokes equations. To ensure stability of the discrete variational problem, we apply ideas from the variational multi-scale method. The finite element problem is posed on the ``full" space-time domain, considering time as another dimension. We provide a rigorous analysis of the stability and convergence of the stabilized formulation. And finally, we apply this method on two benchmark problems in computational fluid dynamics, namely, lid-driven cavity flow and flow past a circular cylinder. We validate the current method with existing results from literature and show that very large space-time blocks can be solved using our approach.

math.NA↗

arXiv2022

Neural PDE Solvers for Irregular Domains

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically address the imposition of Dirichlet/Neumann boundary conditions over irregular domain boundaries. In this paper, we present a framework to neurally solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Our network takes in the shape of the domain as an input (represented using an unstructured point cloud, or any other parametric representation such as Non-Uniform Rational B-Splines) and is able to generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a novel approach for identifying the interior and exterior of the computational grid in a differentiable manner. We also perform a careful error analysis which reveals theoretical insights into several sources of error incurred in the model-building process. Finally, we showcase a wide variety of applications, along with favorable comparisons with ground truth solutions.

cs.LG↗

arXiv2021

NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs

We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ collocation-based methods to make point-wise predictions of solutions to PDEs. This approach has the advantage of naturally enforcing different boundary conditions as well as ease of invoking well-developed PDE theory -- including analysis of numerical stability and convergence -- to obtain capacity bounds for our proposed neural networks in discretized domains. We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkin loss function based on the Finite Element Method (FEM) on a parametric elliptic PDE. The weighted Galerkin loss (FEM loss) is similar to an energy functional that produces improved solutions, satisfies a priori mesh convergence, and can model Dirichlet and Neumann boundary conditions. We prove theoretically, and illustrate with experiments, convergence results analogous to mesh convergence analysis deployed in finite element solutions to PDEs. These results suggest that a mesh-based neural network approach serves as a promising approach for solving parametric PDEs with theoretical bounds.

cs.LG↗

arXiv2021

Differentiable Spline Approximations

The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable "layer" in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis.

cs.LG↗

arXiv2021

Distributed Multigrid Neural Solvers on Megavoxel Domains

We consider the distributed training of large-scale neural networks that serve as PDE solvers producing full field outputs. We specifically consider neural solvers for the generalized 3D Poisson equation over megavoxel domains. A scalable framework is presented that integrates two distinct advances. First, we accelerate training a large model via a method analogous to the multigrid technique used in numerical linear algebra. Here, the network is trained using a hierarchy of increasing resolution inputs in sequence, analogous to the 'V', 'W', 'F', and 'Half-V' cycles used in multigrid approaches. In conjunction with the multi-grid approach, we implement a distributed deep learning framework which significantly reduces the time to solve. We show the scalability of this approach on both GPU (Azure VMs on Cloud) and CPU clusters (PSC Bridges2). This approach is deployed to train a generalized 3D Poisson solver that scales well to predict output full-field solutions up to the resolution of 512x512x512 for a high dimensional family of inputs.

cs.LG↗

arXiv2021

Industrial scale large eddy simulations (LES) with adaptive octree meshes using immersogeometric analysis

We present a variant of the immersed boundary method integrated with octree meshes for highly efficient and accurate Large-Eddy Simulations (LES) of flows around complex geometries. We demonstrate the scalability of the proposed method up to $\mathcal{O}(32K)$ processors. This is achieved by (a) rapid in-out tests; (b) adaptive quadrature for an accurate evaluation of forces; (c) tensorized evaluation during matrix assembly. We showcase this method on two non-trivial applications: accurately computing the drag coefficient of a sphere across Reynolds numbers $1-10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency.

math.NA↗

arXiv2020

Deep Generative Models that Solve PDEs: Distributed Computing for Training Large Data-Free Models

Recent progress in scientific machine learning (SciML) has opened up the possibility of training novel neural network architectures that solve complex partial differential equations (PDEs). Several (nearly data free) approaches have been recently reported that successfully solve PDEs, with examples including deep feed forward networks, generative networks, and deep encoder-decoder networks. However, practical adoption of these approaches is limited by the difficulty in training these models, especially to make predictions at large output resolutions ($\geq 1024 \times 1024$). Here we report on a software framework for data parallel distributed deep learning that resolves the twin challenges of training these large SciML models - training in reasonable time as well as distributing the storage requirements. Our framework provides several out of the box functionality including (a) loss integrity independent of number of processes, (b) synchronized batch normalization, and (c) distributed higher-order optimization methods. We show excellent scalability of this framework on both cloud as well as HPC clusters, and report on the interplay between bandwidth, network topology and bare metal vs cloud. We deploy this approach to train generative models of sizes hitherto not possible, showing that neural PDE solvers can be viably trained for practical applications. We also demonstrate that distributed higher-order optimization methods are $2-3\times$ faster than stochastic gradient-based methods and provide minimal convergence drift with higher batch-size.

cs.LG↗