Applications

Chronos has been used in several applications arising arising from challenging real-world problems. These set of problems can be grouped into two classes denoted as Fluid dynamic and Mechanical. The one consists of a series of problems arising from the discretization of the Laplace operator and related to fluid dynamic problems, such as underground fluid flow (reservoir), compressible or incompressible airflow around complex geometries (CFD) or porous flow (porous flow). The second category includes problems related to mechanical applications such as subsidence analysis, hydrocarbon recovery, gas storage (geomechanics), mesoscale simulation of composite materials (mesoscale), mechanical deformation of human tis- sues or organs subjected to medical interventions (biomedicine), design and analysis of mechanical elements, e.g., cutters, gears, air-coolers (mechanical).

Test case finger4m

The matrix finger4m derives from a two-dimensional Darcy flow of a binary mixture. The physical model describes flow in a porous medium or Hele-Shaw cell, a thin gap between two parallel plates. The behavior of the system, and hence the matrix, is governed by two nondimensional groups: the P'eclet number \(Pe = 10^4\) and the viscosity ratio \(M=\exp(3.5)\).

Test cases guenda11m and geo61m

The matrices guenda11m and geo61m derive from two 3D geomechanical models of a reservoir.

In particular, the matrix guenda11m derives from a domain that spans an area of \(40 \times 40 \, km^{2}\) and extends down to \(5 \, km\) depth. To reproduce with high fidelity the real geometry of the gas reservoir, a severely distorted mesh with \(22,665,896\) linear tetrahedra and \(3,817,466\) vertices is used. While fixed boundaries are prescribed on the bottom and lateral sides, the surface is traction-free. The matrix geo61m represents a geological formation with 479 layers. The geometry of the modeled domain is characterized by an area of \(55 \times 40 \, km^{2}\) with the reservoir in an almost barycentric position and the base at a depth of \(6.5 \, km\). The grid is based on a mesh of \(20,354,736\) brick elements. The Figure below shows a representation of the problem’s geometry and mesh. As can be observed, some elements are highly distorted to reproduce the geological layers.

Test case agg14m

The mesh derives from a 3D mesoscale simulation of an heterogeneous cube of lightened concrete. The domain has dimensions \(50 \times 50 \times 50 \, mm^{3}\) and contains \(2644\) spherical inclusions of polystyrene. The cement matrix is characterized by \((E_1, \nu_1) = (25,000 MPa, 0.30)\), while the polystyrene inclusions are characterized by \((E_2, \nu_2) = (5 MPa, 0.30)\). Hence, the contrast between the Young modules of these two linear elastic materials is extremely high. The discretization is done via tetrahedral finite elements. The Figure below shows a representation of the problem’s geometry and mesh.

Test case M20

The mesh derives from the 3D mechanical equilibrium of a symmetric machine cutter that is loosely constrained. The unstructured mesh is composed by \(4,577,974\) second order tetrahedra and \(6,713,144\) vertices resulting in \(20,056,050\) DOFs. Material is linear elastic with \((E,\nu) = (10^{8} MPa, 0.33)\). This problem was initially presented by and later used in the work.

Test case tripod24m

The mesh derives from the 3D mechanical equilibrium of a tripod with clamped bases. Material is linear elastic with \((E,\nu) = (10^{6} MPa, 0.45)\). The mesh is formed by linear tetrahedra and discretization is given by the finite element method. Figure below shows the geometry and the mesh of the problem.

Test case poi65m

The mesh derives from the solution of the Poisson’s equation \(\nabla^2 \phi = f\) over a 3D cube. The domain is discretized with a \(100 \times 200 \times 402\) finite difference grid.

Test case Pflow73m

The mesh derives from a basin model, with the discretization of a \(178.8 \times 262.0 \, km^{2}\) geological area - at the end of basin evolution - with a mesh of 20-node hexahedral elements. The Pflow73m matrix derives from the discretization of the mass conservation and Darcy’s law. Due to strong permeability contrasts between neighboring elements and geometrical distortion of the computational grid, the matrix is severely ill-conditioned and challenging to solve.

Test case c4zz134m

The mesh derives from the discrtization of the complex conformation of the urethral duct, with particular regard to the bulbar region. The duct locally consists of an inner thin layer of dense connective tissue and an outer thick stratum of more compliant spongy tissue. Both the materials are linear elastic, characterized by \((E,\nu) = (0.06 MPa, 0.4)\) and \((E,\nu) = (0.0066 MPa, 0.4)\), respectively. The Figure below shows a representation of the problem’s geometry and mesh.