Andrea Franceschini

Andrea Franceschini, Matteo Frigo, Carlo Janna et al.

Frictional contact is one of the most challenging problems in computational mechanics. Typically, it is a tough nonlinear problem often requiring several Newton iterations to converge and causing troubles also in the solution to the related linear systems. When contact is modeled with the aid of Lagrange multipliers, the impenetrability condition is enforced exactly, but the associated Jacobian matrix is indefinite and needs a special treatment for a fast numerical solution. In this work, a constraint preconditioner is proposed where the primal Schur complement is computed after augmenting the zero block. The name Reverse is used in contrast to the traditional approach where only the structural block undergoes an augmentation. Besides being able to address problems characterized by singular structural blocks, often arising in contact mechanics, it is shown that the proposed approach is significantly cheaper than traditional constraint preconditioning for this class of problems and it is suitable for an efficient HPC implementation through the Chronos parallel package. Our conclusions are supported by several numerical experiments on mid- and large-size problems from various applications. The source files implementing the proposed algorithm are freely available on GitHub.

Andrea Franceschini, Laura Gazzola, Massimiliano Ferronato

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. The porous medium is discretized using low-order continuous finite elements, with cell-centered Lagrange multipliers and pressure unknowns used to impose the constraints and solve the fluid flow in the fractures, respectively. This formulation does not require any interpolation between different fields, but is not uniformly inf-sup stable and requires a stabilization. For the resulting 3 x 3 block Jacobian matrix, we design scalable preconditioning strategies, based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid preconditioners. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a real-world problem.

Andrea Franceschini, Victor A. Paludetto Magri, Gianluca Mazzucco et al.

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.

Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato

The growing availability of computational resources has significantly increased the interest of the scientific community in performing complex multi-physics and multi-domain simulations. However, the generation of appropriate computational grids for such problems often remains one of the main bottlenecks. The use of a domain partitioning with non-conforming grids is a possible solution, which, however, requires the development of robust and efficient inter-grid interpolation operators to transfer a scalar or a vector field from one domain to another. This work presents a novel approach for interpolating quantities across non-conforming meshes within the framework of the classical mortar method, where weak continuity conditions are enforced. The key contribution is the introduction of a novel strategy that uses mesh-free Radial Basis Function (RBF) interpolations to compute the mortar integral, offering a compelling alternative to traditional projection-based methods. We propose an efficient algorithm tailored for complex three-dimensional settings allowing for potentially significant savings in the overall computational cost and ease of implementation, with no detrimental effects on the numerical accuracy. The formulation, analysis, and validation of the proposed RBF-based algorithm is discussed with the aid of a set of numerical examples, demonstrating its effectiveness. Furthermore, the details of the implementation are discussed and a test case involving a complex geometry is presented, to illustrate the applicability and advantages of our approach in real-world problems.

Carlo Janna, Andrea Franceschini, Jacob B. Schroder et al.

Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches.

Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato

Accurate numerical simulation of fault and fracture mechanics is critical for the performance and safety assessment of many subsurface systems. The discretized representation of discontinuity surfaces and the robust simulation of their frictional contact behavior still represent major challenges. In this work, we use the mortar method to enforce the contact constraints and allow for non-conformity around the discontinuity surface, with a set of Lagrange multipliers playing the role of interface tractions. The formulation combines piecewise linear displacements with piecewise constant multipliers defined on one side of the fault interface (the non-mortar side). This choice for the Lagrange multipliers has a number of advantages from practical and computational viewpoints, but violates the inf-sup stability constraint. In order to stabilize the proposed formulation, we develop a traction-jump stabilization term to be added to the constraint equations. We use a macro-element analysis to derive an algorithmic strategy that automatically evaluates the proper scaling of the stabilization, without requiring any additional user-selected parameter. Numerical experiments demonstrate that the proposed formulation not only restores the inf-sup stability condition, but also recovers stable traction profiles in the presence of finer non-mortar sides, where other inf-sup-stable formulations fail. The proposed method is finally used to simulate non-linear contact conditions in non-conforming corner-point grids typically used in industrial geological applications.