Research Papers

Real-Time simulations are becoming increasingly common for various applications, from geometric design to medical simulation. In a series of papers, we developed real-time simulation of the interaction of a surgeon or interventional radiologist with deformable organs. Such simulations are useful to both, help surgeons train, rehearse complex operations or/and to guide them during the intervention. In time, reliable simulations could also be central to robotic surgery.

Our approach is the first a posteriori error-driven adaptive finite element for real-time simulation, and we demonstrated its practical usability on a percutaneous procedure involving needle insertion in a liver, as well as the simulation of electrode implantation for deep brain stimulation. We take into account the brain shift phenomena occurring when a craniotomy is performed. Through academic and practical examples we demonstrate that our adaptive approach, as compared with a uniform coarse mesh, increases the accuracy of the displacement and stress fields around the needle shaft and, for a given accuracy, saves computational time with respect to a uniform finer mesh. This facilitates real-time simulations. The proposed methodology has direct implications in increasing the accuracy, and controlling the computational expense of the simulation of percutaneous procedures such as biopsy, brachytherapy, regional anaesthesia, or cryotherapy. Moreover, the proposed approach can be helpful in the development of robotic surgeries because the simulation taking place in the control loop of a robot needs to be accurate, and to occur in real time.

Fretting is a phenomenon that occurs at the contacts of surfaces that are subjected to oscillatory relative movement of small amplitudes, for example, railway tracks or pistons in automobile engines. Depending on service conditions, fretting may significantly reduce the service life of a component due to fretting fatigue. In this regard, the analysis of stresses at contact is of great importance for predicting the lifetime of components.

We constructed finite element models, with different element sizes, in order to verify the existence of stress singularity under fretting conditions. Our analyses showed that the convergence rate in stress components depends on coefficient of friction, implying that this rate also depends on the loading condition. It was also observed that errors can be relatively high for cases with a high coefficient of friction, suggesting the importance of mesh refinement in these situations. We provided some recommendations of mesh sizes to perform FE analysis of fretting problems, with different levels of accuracy.

The maritime logistics and service market size stood at ~USD 83 Billion in 2023, and is expected to grow ~USD 150 Billion by 2032. Clearly, developing good vessels, which can withstand harsh water conditions is an important research area. Numerical computations of free-surface waves and their impact on solid bodies require a higher order accurate discretization, both in space and time, in order to obtain accurate solutions with minimal dispersion and dissipation errors, in particular for long time simulations.

We developed two classes of discontinuous Galerkin methods where the free-surface boundary conditions are included into the weak formulation or treated separately. When the free-surface boundary conditions were included into the weak formulation, the L2-norm error in the wave height and velocity potential of the numerical solution is of optimal order, without the need for a separate velocity reconstruction. When the free-surface boundary conditions were treated separately, this approach results in a uniformly higher order accurate finite element discretization, which we demonstrated for linear free-surface waves in an inviscid incompressible fluid. The study of this model problem is an essential step in the development of a finite element method for nonlinear waves.

To obtain reliable numerical solution of any real-world problem using computers, there are 4 steps involved: (1) Modelling the physical phenomenon into mathematical equations, (2) Discretizing (representing) the mathematical equations on finite number of points of the domain of interest with suitable numerical methods, which results in a large linear system of equations to be solved on computers, (3) Solving the large linear system of equations on computers, typically using iterative solution methods, and (4) Computing accurate a-posteriori error estimates for the numerical solution. In this research topic, we are concerned with step (4).

In a series of papers, we derived functional-type two-sided guaranteed and sharp a-posteriori error bounds for various class of problems, namely, discontinuous Galerkin (DG) discretisation of elliptic problems, space-conforming and nonconforming discretisation of time-dependent convection-diffusion problems, nonconforming approximations of elliptic problems, and isogeometric discretisation of elliptic problems.

Moreover, for surgical simulation, which is an extremely important cutting-edge research topic, we also utilized a different class of a-posteriori estimates, and another class of a-posteriori estimates in studies involving plate vibrations.

List of papers: (2, 5, 7, 8), 10, 11, 17, 18, 20, 21.

To obtain reliable numerical solution of any real-world problem using computers, there are 4 steps involved: (1) Modelling the physical phenomenon into mathematical equations, (2) Discretizing (representing) the mathematical equations on finite number of points of the domain of interest with suitable numerical methods, which results in a large linear system of equations to be solved on computers, (3) Solving the large linear system of equations on computers, typically using iterative solution methods, and (4) Computing accurate a-posteriori error estimates for the numerical solution. In this research topic, we are concerned with step (3).

The standard direct methods to solve linear systems of equations have a computational complexity of O(N3), whereas the state of the art optimal order iterative methods typically have a computational complexity of O(N), where N denotes the number of unknowns. This results in a huge saving in computational speed and memory requirements. In a series of papers, we developed optimal order iterative solvers for discontinuous Galerkin (DG) discretization of elliptic problems in 2d and 3d (including anisotropic case), for conforming finite element discretization in H(curl) and H(div) spaces, and for the discretization based on isogeometric analysis approach.

For the test problems based on DG discretization, the problem description and some results are presented in Pictures 1-5. For a test problem in H(curl) and H(div) spaces, the representative results are presented in Picture 6. When comparing the results obtained from our algorithm (the second-last column) against the results obtained by using the default Matlab solver (the last column), it is clear that our algorithm is approximately 100 times faster for a moderate problem size of about 6.3 million unknowns. Moreover, this difference in solution time will continue to grow with the problem size because our algorithm is an order of N faster than the best algorithm used by the default Matlab solver.

For geometries with smooth boundaries, the elliptic problems can be solved to exponential accuracy using standard finite element methods. However, they become particularly challenging in the presence of sharp corners or kinks in the geometry. The method we proposed is exponentially accurate, and of almost optimal order. The algorithm was implemented on a distributed memory parallel computer.