arXiv subjects

Johannes L. Grafen

Publications and source records attributed to Johannes L. Grafen.

3 recordsLinked to original sources

arXiv2026

Benchmark cases for gradient-based optimization in linear elastic solid mechanics using lattice Boltzmann methods

Gradient-based optimization (GBO) of problems constrained by partial differential equations is central to parameter identification and structural design in solid mechanics. Lattice Boltzmann method (LBM) schemes for linear elastic solids have only recently been derived. Combined with automatically generated discrete adjoint collision kernels, they make GBO possible with a purely LBM-based primal and adjoint solver. We propose, to our knowledge, the first optimization benchmarks for solid LBM: three inverse problems, each with a manufactured solution of the forward problem and a known reference control. These are the identification of two scalar amplitudes of a body force field on a periodic domain, the distributed control of the same field with up to $12800$ degrees of freedom, and the identification of Young's modulus on an elliptic plate with Dirichlet and Neumann boundaries. The primal scheme converges with second order on the periodic domain and with first order on the elliptic plate. The recovered controls follow these rates, with experimental orders of convergence of $1.92$ and $1.06$, respectively. Gradients obtained via forward-mode automatic differentiation (f-AD) and the discrete adjoint LBM (ALBM) require the same number of optimization steps and yield control errors agreeing to at least eleven significant digits. The ALBM gradient is further validated node-wise against a finite difference quotient, with deviations decreasing under mesh refinement. The benchmarks are implemented in the open-source LBM framework OpenLB and provide a verification basis for optimization in LBM-based solid mechanics and its extension to fluid-structure interaction.

math.OC↗

arXiv2026

Computing statistical Euler limits of the Navier--Stokes equations in three dimensions

We develop a Monte Carlo lattice Boltzmann method to compute statistical solutions to the three-dimensional incompressible Navier--Stokes and Euler equations. Entropic space-time adaptive relaxation of the higher order kinetic moments yields stable numerical solutions with decreasing viscosity. We provide a convergence analysis that is conditional on four explicitly stated assumptions regarding the discrete dynamics. Under diffusive scaling, the laws of the discrete ensemble converge along a subsequence to a limit satisfying the Foias--Temam Liouville formulation of the Navier--Stokes equations. Consequently, provided the structure function scaling holds uniformly, the vanishing viscosity limit of these measures satisfies the multi-point statistical Euler hierarchy of Fjordholm, Mishra, and Weber. The limit measures inherit the known weak-strong uniqueness principle on the interval of existence of a strong Euler solution. Under explicit scaling assumptions, a Kuznetsov-type argument yields a fractional 1-Wasserstein convergence rate. We present three-dimensional computations of time-dependent statistical solutions along the inviscid limit of the incompressible Navier--Stokes equations together with convergence measurements in the Wasserstein metric. Numerical experiments on a randomized Taylor--Green vortex with 24-dimensional initial uncertainty recover Kolmogorov's K41 scaling for energy spectra and structure functions, exhibit the failure of pathwise strong convergence, and yield Wasserstein convergence rates of about $0.5$ at the onset of turbulence. Finally, error measurements with respect to spectral hyperviscosity computations indicate that the computed limit measure is independent of the numerical regularization.

math.NA↗

arXiv2026

Mean-Flow Adjoint Sensitivity Analysis of Unsteady Flow Around Porous Cylinders Using a Homogenized Lattice Boltzmann Method

Adjoint-based sensitivity analysis is an indispensable tool for large-scale fluid-dynamic design and distributed control problems, yet its application to unsteady and turbulent flows is frequently hindered by the prohibitive memory footprint of transient checkpointing and the divergence of gradients in chaotic regimes. To address these computational bottlenecks, this paper presents a mean-flow adjoint sensitivity analysis framework for unsteady flows around porous cylinders using the homogenized lattice Boltzmann method (HLBM). Within this framework, solid structures are efficiently modeled as local porous media utilizing a Brinkman penalization approach. We systematically investigate HLBM-based adjoint gradients for drag and energy dissipation objective functionals, transitioning from steady laminar to unsteady, and finally to turbulent flow regimes. For the turbulent case at Re = 3900, a proof-of-concept is conducted where the framework relies on automatic differentiation to automatically generate adjoint kernels containing subgrid-scale (SGS) turbulence models for large eddy simulations (LES), circumventing manual derivation and allowing for a direct comparison against the frozen turbulence assumption (FTA).

physics.flu-dyn↗