Computer Science

Numerical computation of the Karhunen--Loève expansion is computationally challenging in terms of both memory requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the K--L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed by the authors in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2]. Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth covariance kernels, while the collocation method performs slightly better for rough covariance kernels.

Mixtures of fluids and granular sediments play an important role in many industrial, geotechnical, and aerospace engineering problems, from waste management and transportation (liquid--sediment mixtures) to dust kick-up below helicopter rotors (gas--sediment mixtures). These mixed flows often involve bulk motion of hundreds of billions of individual sediment particles and can contain both highly turbulent regions and static, non-flowing regions. This breadth of phenomena necessitates the use of continuum simulation methods, such as the material point method (MPM), which can accurately capture these large deformations while also tracking the Lagrangian features of the flow (e.g.\ the granular surface, elastic stress, etc.). Recent works using two-phase MPM frameworks to simulate these mixtures have shown substantial promise; however, these approaches are hindered by the numerical limitations of MPM when simulating pure fluids. In addition to the well-known particle ringing instability and difficulty defining inflow/outflow boundary conditions, MPM has a tendency to accumulate quadrature errors as materials deform, increasing the rate of overall error growth as simulations progress. In this work, we present an improved, two-phase continuum simulation framework that uses the finite volume method (FVM) to solve the fluid phase equations of motion and MPM to solve the solid phase equations of motion, substantially reducing the effect of these errors and providing better accuracy and stability for long-duration simulations of these mixtures.

Neural networks with sufficiently smooth activation functions can approximate values and derivatives of any smooth function, and they are differentiable themselves. We improve the approximation capability of neural networks by utilizing the differentiability of neural networks; the gradient and Hessian of neural networks are used to train the neural networks to satisfy the differential equations of the problems of interest. Several activation functions are also compared in term of effective differentiation of neural networks. We apply the differential neural networks to the pricing of financial options, where stochastic differential equations and the Black-Scholes partial differential equation represent the relation of price of option and underlying assets, and the first and second derivatives, Greeks, of option play important roles in financial engineering. The proposed neural network learns -- (a) the sample paths of option prices generated by stochastic differential equations and (b) the Black-Scholes equation at each time and asset price. Option pricing experiments were performed on multi-asset options such as exchange and basket options. Experimental results show that the proposed method gives accurate option values and Greeks; sufficiently smooth activation functions and the constraint of Black-Scholes equation contribute significantly for accurate option pricing.

Beam finite elements for non linear plastic analysis of beam-like structures are formulated according to Displacement Based (DB) or Force Based (FB) approaches. DB formulations rely on modelling the displacement field by means of displacement shape functions. Despite the greater simplicity of DB over FB approaches, the latter provide more accurate responses requiring a coarser mesh. To fill the existent gap between the two approaches the improvement of the DB formulation without the introduction of mesh refinement is needed. To this aim the authors recently provided a contribution to the improvement of the DB approach by proposing new enriched adaptive displacement shape functions leading to the Smart Displacement Based (SDB) beam element. In this paper the SDB element is extended to include the axial force-bending moment interaction, crucial for the analysis of r/c cross sections. The proposed extension requires the formulation of discontinuous axial displacement shape functions dependent on the diffusion of plastic deformations. The stiffness matrix of the extended smart element is provided explicitly and shown to be dependent on the displacement shape functions updating. The axial force-bending moment interaction is approached by means of a fibre discretisation of the r/c cross section. The extended element, addressed as Fibre Smart Displacement Based (FSDB) beam element, is shown to be accurate and furthermore accompanied by the proposal of an optional procedure to verify strong equilibrium of the axial force along the beam element, which is usually not accomplished by DB beam elements. Given a fixed mesh discretisation the performance of the FSDB beam element is compared with the DB approach to show the better accuracy of the proposed element.

Finite element model updating of a structure made of linear elastic materials is based on the solution of a minimization problem. The goal is to find some unknown parameters of the finite element model (elastic moduli, mass densities, constraints and boundary conditions) that minimize an objective function which evaluates the discrepancy between experimental and numerical dynamic properties. The objective function depends nonlinearly on the parameters and may have multiple local minimum points. This paper presents a numerical method able to find a global minimum point and assess its reliability. The numerical method has been tested on two simulated examples - a masonry tower and a domed temple - and validated via a generic genetic algorithm and a global sensitivity analysis tool. A real case study monitored under operational conditions has also been addressed, and the structure's experimental modal properties have been used in the model updating procedure to estimate the mechanical properties of its constituent materials.

We present a theoretical framework to model the electric response of cell aggregates. We establish a coarse representation for each cell as a combination of membrane and cytoplasm dipole moments. Then we compute the effective conductivity of the resulting system, and thereafter derive a Fokker-Planck partial differential equation that captures the time-dependent evolution of the distribution of induced cellular polarizations in an ensemble of cells. Our model predicts that the polarization density parallel to an applied pulse follows a skewed t-distribution, while the transverse polarization density follows a symmetric t-distribution, which are in accordance with our direct numerical simulations. Furthermore, we report a reduced order model described by a coupled pair of ordinary differential equations that reproduces the average and the variance of induced dipole moments in the aggregate. We extend our proposed formulation by considering fractional order time derivatives that we find necessary to explain anomalous relaxation phenomena observed in experiments as well as our direct numerical simulations. Owing to its time-domain formulation, our framework can be easily used to consider nonlinear membrane effects or intercellular couplings that arise in several scientific, medical and technological applications.

A novel finite element framework is proposed for the numerical simulation of two phase flows with surface tension. The Level-Set (LS) method with piece-wise quadratic (P2) interpolation for the liquid-gas interface is used in order to reach higher-order convergence rates in regions with smooth interface. A balanced-force implementation of the continuum surface force model is used to take into account the surface tension and to solve static problems as accurately as possible. Given that this requires a balance between the discretization used for the LS function, and that used for the pressure field, an equal-order P2/P2/P2 scheme is proposed for the Navier-Stokes and LS advection equations, which are strongly coupled with each other. This fully implicit formulation is stabilized using the residual-based variational multiscale framework. In order to improve the accuracy and obtain optimal convergence rates with a minimum number of elements, an anisotropic mesh adaption method is proposed where the unstructured mesh is kept as fine as possible close to the zero iso-value of the P2 LS function. Elements are automatically stretched in regions with flat interface in order to keep the complexity fixed during the simulation. The accuracy and efficiency of this approach are demonstrated for two and three dimensional simulations of a rising bubble.

Capturing the interaction between objects that have an extreme difference in Young s modulus or geometrical scale is a highly challenging topic for numerical simulation. One of the fundamental questions is how to build an accurate multi-scale method with optimal computational efficiency. In this work, we develop a material-point-spheropolygon discrete element method (MPM-SDEM). Our approach fully couples the material point method (MPM) and the spheropolygon discrete element method (SDEM) through the exchange of contact force information. It combines the advantage of MPM for accurately simulating elastoplastic continuum materials and the high efficiency of DEM for calculating the Newtonian dynamics of discrete near-rigid objects. The MPM-SDEM framework is demonstrated with an explicit time integration scheme. Its accuracy and efficiency are further analysed against the analytical and experimental data. Results demonstrate this method could accurately capture the contact force and momentum exchange between materials while maintaining favourable computational stability and efficiency. Our framework exhibits great potential in the analysis of multi-scale, multi-physics phenomena

We propose a novel mixed displacement-pressure formulation based on an energy functional that takes into account the relation between the pressure and the volumetric energy function. We demonstrate that the proposed two-field mixed displacement-pressure formulation is not only applicable for nearly and truly incompressible cases but also is consistent in the compressible regime. Furthermore, we prove with analytical derivation and numerical results that the proposed two-field formulation is a simplified and efficient alternative for the three-field displacement-pressure-Jacobian formulation for hyperelastic materials whose strain energy density functions are decomposed into deviatoric and volumetric parts.

The FE 2 homogenization algorithm for multiscale modeling iterates between the macroscale and the microscale (represented by a representative volume element) till convergence is achieved at every increment of macroscale loading. The information exchange between the two scales occurs at the gauss points of the macroscale finite element discretization. The microscale problem is also solved using finite elements on-the-fly thus rendering the algorithm computationally expensive for complex microstructures. We invoke machine learning to establish the input-output causality of the RVE boundary value problem using a neural network framework. This renders the RVE as a blackbox which gets the information from the macroscale as an input and gives information back to the macroscale as output, thereby eliminating the need for on-the-fly finite element solves at the RVE level. This framework has the potential to significantly accelerate the FE 2 algorithm.