In the dynamic analysis of discretized elastic bodies with contacts/impacts, the common formulations to model contact force include the penalty method and the Lagrangian method, which are different in the constraint imposition strategies. Traditionally, the Lagrangian method is thought to be less efficient due to additional multipliers and numerical complexity, however, the viewpoint is challenged in this paper. The goal of this paper is to investigate how numerical efficiency and accuracy using the two different methods are influenced by some non-physical chosen parameters such as stiffness coefficient, time step size and spatial discretization. An experimental sphere-rod impact problem and a multi-point impact problem are solved to evaluate certain numerical intricacies of the two implementations. The results show that in dealing with normal impact problems, the choice of penalty factor and time step size using the penalty method is not an omissible act to obtain accuracy and stability, while there are no such manually-defined parameters using the Lagrangian method. Moreover, there seems to be a clear advantage of the Lagrangian method in which much less mesh elements are needed to achieve the same accuracy compared to those of the penalty method.

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.

An accurate cartesian method is devised to simulate incompressible viscous flows past an arbitrary moving body. The Navier-Stokes equations are spatially discretized onto a fixed Cartesian mesh. The body is taken into account via the ghost-cell method and the so-called penalty method, resulting in second-order accuracy in velocity. The accuracy and the efficiency of the solver are tested through two-dimensional reference simulations. To show the versatility of this scheme we simulate a three-dimensional self propelled jellyfish prototype.

This paper investigates the eigenmode optimization problem governed by the scalar Helmholtz equation in continuum system in which the computed eigenmode approaches the prescribed eigenmode in the whole domain. The first variation for the eigenmode optimization problem is evaluated by the quadratic penalty method, the adjoint variable method, and the formula based on sensitivity analysis. A penalty optimization algorithm is proposed, in which the density evolution is accomplished by introducing an artificial time term and solving an additional ordinary differential equation. The validity of the presented algorithm is confirmed by numerical results of the first and second eigenmode optimizations in 1D and 2D problems.

A theoretical investigation of the unsteady flow of a Newtonian fluid through a channel is presented using an alternative boundary condition to the standard no-slip condition, namely the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter called the slip length, and the most general case of a constant but different slip length on each channel wall is studied. An analytical solution for the velocity distribution through the channel is obtained via a Fourier series, and is used as a benchmark for numerical simulations performed utilizing a finite element analysis modified with a penalty method to implement the slip boundary condition. Comparison between the analytical and numerical solution shows excellent agreement for all combinations of slip lengths considered.

We study the partial differential equation

        max{Lu − f, H(Du)} = 0

where u is the unknownfunction, L is a second-order elliptic operator, f is agiven smooth function and H is a convex function. This is a modelequation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problem that has aHölder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming ina viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity andopposite direction applied on the rigid bacterial body and on an associated region in thefluid representing the flagellar bundle. The method for solving the fluid flow and themotion of the bacteria is based on a variational formulation written on the whole domain,strongly coupling the fluid and the rigid particle problems: rigid motion is enforced bypenalizing the strain rate tensor on the rigid domain, while incompressibility is treatedby duality. This model allows to achieve an accurate description of fluid motion andhydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopicmodel is also used, in which the size of the bacteria is supposed to be much smaller thanthe elements of fluid: the perturbation of the fluid due to propulsion and motion of theswimmers is neglected, and the fluid is only subjected to the buoyant forcing induced bythe presence of the bacteria, which are denser than the fluid. Although this model doesnot accurately take into account hydrodynamic interactions, it is able to reproducecomplex collective dynamics observed in concentrated bacterial suspensions, such asbioconvection. From a mathematical point of view, both models lead to a minimizationproblem which is solved with a standard Finite Element Method. In order to ensurerobustness, a projection algorithm is used to deal with contacts between particles. Wealso reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation onthe oxygen concentration is solved in the fluid domain, with a source term accounting foroxygen consumption by the bacteria. The orientations of the individual bacteria aresubjected to random changes, with a frequency that depends on the surrounding oxygenconcentration, in order to favor the direction of the concentration gradient.