Mesh refinement based on error estimation

Our software is a cloud-based computational tool that automates the entire process of flow simulation by building the model without any human assistance.

The entire simulation process is automated by Ingrid Cloud proprietary, self-learning algorithms. Full computerization increases accuracy and eliminates costly human errors. The optimized algorithms also reduce computing costs by reducing the computational resources necessary to run a simulation.

The foundation

Ingrid Cloud is built on a ground-breaking Computational Fluid Dynamics (CFD) framework, which uses the Finite Element Method (FEM) together with adaptive mesh refinement based on adjoint techniques and a posteriori error estimation.

Since 2010, Ingrid Cloud’s technology has been regularly validated in benchmark workshops organized by the American Institute of Aeronautics and Astronautics (AIAA), by NASA and by other institutes. The Ingrid Cloud team has dedicated over 30 person-years of research to:

  • Develop a methodology that allows the automatic creation of highly-complex simulation models from CAD-geometries.
  • Implement this method in a scientific code.
  • Optimize and test the code on the fastest supercomputers in the world (K computer in Japan, Beskow in Stockholm, and Hazel Hen in Stuttgart, among others).

Unique characteristics

  • High fidelity turbulent flow simulations.
  • Setup simulation with simple wizard and intuitive user interface.
  • Automated, smart mesh generation based on adjoint techniques and error estimation.
  • No explicit turbulence model needed.
  • Very few input parameters when compared to traditional CFD.


The technology is based on two unique innovations that provide superior capabilities. The method resulting from these innovations has no adjustable parameters, and no ad-hoc design of the mesh is needed, enabling full automation of the simulation workflow.

Ingrid Cloud framework capabilities:

  • A parameter-free method for simulation of turbulent flow at high Reynolds numbers, in the form of weak solutions of the Navier–Stokes equations approximated by adaptive FEM. Here, viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and the effect of skin-friction is observed to be small, compared to inertial effects.
  • Algorithms for adaptive optimization of the mesh based on adjoint techniques and a posteriori error estimation, so that the output quantities of interest, in the form of functionals of the solution, converge to become independent of mesh resolution


  • Hoffman, J. (2005), Computation of Mean Drag for Bluff Body Problems Using Adaptive DNS/LES, SIAM Journal on Scientific Computing, 27:1, 184-207.
  • Hoffman, J. (2006), Adaptive simulation of the subcritical flow past a sphere. Journal of Fluid Mechanics, 568, 77-88. doi:10.1017/S0022112006002679
  • Hoffman, J. (2009), Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2, Int. J. Numer. Meth. Fluids, 59: 1241–1258. doi:10.1002/fld.1865.
  • Jansson, N., Hoffman, J, Jansson, J. (2012), Framework for Massively Parallel Adaptive Finite Element Computational Fluid Dynamics On Tetrahedral Meshes, SIAM Journal on Scientific Computing, vol. 34, no. 1, s. C24-C42.
  • Vilela de Abreu, R., Jansson, N., Hoffman, J (2016), Computation of aeroacoustic sources for a Gulfstream G550 nose landing gear model using adaptive FEM, Computers & Fluids, 124: 136-146, ISSN 0045-7930,