Molecular Dynamics as a Mathematical Mapping. III. Efficient Evaluation of the Differentiable Force Functions and Their Derivatives

Application of holography to anisotropic composite plates

A computational algorithm based on the multiquadric, which is a continuously differentiable radial basis function, is devised to solve the shallow water equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from partial derivatives of the radial basis function, not by any difference scheme. The method does not require the generation of a grid as in the finite-element method and allows easy editing and refinement of the numerical model. To increase confidence in the multiquadric solution, a sensitivity and convergence analysis is performed using numerical models of a rectangular channel. Applications of the algorithm are made to compute the sea surface elevations and currents in Tolo Harbour, Hong Kong, during a typhoon attack. The numerical solution is shown to be robust and stable. The computed results are compared with measured data and good agreement is indicated.

The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series that transform the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The resulting math model is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton’s principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition–based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models.

Two continuous-time mathematical computing methods are proposed for solving the multidimensional wave equation leading to realizable analog computing circuits. The proposed analog computing processors will potentially be able to solve a certain special classes of computational problems involving partial differential equations, which are defined from continuous-time systems. The new analog computing methods are first derived and physically implemented for the first-time using low-frequency operational amplifier circuits in order to experimentally verify the correctness of the proposed methods. Both algorithms approximate the spatial domain partial derivatives using discrete finite differences. The first method performs a direct Laplace transform (with respect to the time variable) on the resulting expression. The second method applies the finite difference along the time dimension and then replaces the discrete time difference with a continuous-time delay operator, which in turn, can be realized as an analog all-pass filter. Analog circuit architectures are introduced for different boundary conditions relevant to common electromagnetic simulation problems. A low frequency prototype of the analog wave equation solver (based on method 1) has been designed, realized and tested using board-level operational amplifier circuits. Test results and measurements are provided to demonstrate the wave propagation in the space-time domain.

Chebyshev collocation spectral lattice Boltzmann method in generalized curvilinear coordinates

Retinal Makes Concerted Conformational Changes During Early Stages of Rhodopsin Activation

In this paper, we introduce a two-dimensional (2-D) Node-based Meshless Radial Point Interpolation Method (RPIM) for modeling nonmagnetized plasma. The problem domain is firstly discretized by a set of electric field nodes (E-nodes) and a set of magnetic field nodes (H-nodes). E-nodes and H-nodes are defined in a manner similar to the point-matched time-domain finite-element method. With the RPIM scheme, the field variable at the node is interpolated by a scalar expansion function which combines radial basis function with monomial basis function. After some mathematical manipulations, the shape function of the node in the support domain is obtained. The field variables and corresponding spatial derivatives in field equations can be obtained analytically and the central finite-difference is used to approximate the time derivatives. The leapfrog RPIM meshless method formulations are then obtained. TM wave propagation in isotropic nonmagnetized plasma is simulated to validate the accuracy and efficiency of the proposed method.

Performing molecular dynamics in a fully continuous and differentiable framework can be viewed as a deterministic mathematical mapping between, on one side, the force field parameters that describe the potential energy interactions and input macroscopic conditions, and, on the other, the calculated corresponding macroscopic properties of the bulk molecular system. Within this framework, it is possible to apply standard methods of variational calculus for the computation of the partial derivatives of the molecular dynamics mapping based on the integration of either the adjoint equations or the sensitivity equations of the classical Newtonian equations of motion. We present procedures for these computations in the standard microcanonical (N, V, E) ensemble, and compare the computational efficiency of the two approaches. The general formulations developed are applied to the specific example of bulk ethane fluid. With these procedures in place, it is now possible to compute the partial derivatives of any property determined by molecular dynamics with respect to any input property and any potential parameter. Moreover, these derivatives are computed to essentially the same level of numerical accuracy as the output properties themselves.

An error arises in large eddy simulations when the order of spatial differentiation and filtering operations is commuted on a non-uniform grid. The magnitude of the commutation error is controlled by spatial variations in the width of the filter function, which is conventionally defined in terms of the local grid size. We have previously found large commutation errors in the sub-grid scale (SGS) terms of a numerical dynamo model, particularly near boundaries where the grid size changes most rapidly. In the present study, we propose a correction for the commutation error which is specifically designed for use with the nonlinear gradient SGS model. The commutation error correction for each SGS term can be expressed as the product of the spatial derivative of the second-order moment of the filter function and the second spatial derivative of the SGS term. We test the correction using output from a fully resolved convection-driven dynamo simulation in a rotating plane layer. As an example, we evaluate the SGS heat flux on a coarser grid with and without the commutation error correction. The result is tested using the resolved dynamo solution on a finer grid. We find that the commutation error correction is comparable in magnitude to the SGS heat flux near the boundary. Addition of the correction term significantly improves the modeled SGS heat flux near the boundaries.

We reexamine the interpretation of the normal force measured with a cone-and-plate rheometer for nematic liquid crystals. We point out the fact that the quite widespread belief that the normal force is directly related to the well-known rheological function 𝒩1 (i.e., the first normal stress difference) fails for these complex fluids. After a brief presentation of the theoretical bases leading to the general expression of the normal force, this new approach is applied to nematic liquid crystals within the framework of Leslie-Ericksen theory. In order to avoid heavy numerical computations the full Leslie-Ericksen equations describing the three-dimensional flow within the cone-and-plate cell are reduced, thanks to reasonable approximations, to an effective one-dimensional problem (i.e., a system of partial differential equations with spatial derivatives with respect to one single coordinate). Results for low molecular weight liquid crystals are presented and discussed. The main feature evidenced in this work is the notable difference between the normal force calculated in this way and the 𝒩1 function, in particular the normal force function becomes negative for high shear rate while 𝒩1 stays positive.

Adaptive gridding methods are of fundamental importance both for industry and academia. As one of the computing methods, the Boundary Element Method (BEM) is used to simulate problems whose fundamental solutions are available. The method is usually characterised as constant elements BEM or linear elements BEM depending on the type of interpolation used at the elements. Its popularity is steadily growing because of its advantages over other numerical methods most important of which being that it only involves obtaining data at the boundary and computation of the solution in the domain is merely a case of post processing through the use of an identity. This results in the reduction of the problem dimension by unity. Although there is a reduction in dimension when we use BEM, the method usually results in full matrices which can be expensive to solve. This makes the method costly for problems that require very fine grids like those with hot spots as in the example of impressed current cathodic protection systems (ICCP). The BEM is a global method in nature in that the solution in one node depends on the solutions in all the other nodes of the grid. Hence an error in one node can pollute the solution in all the other nodes. In this thesis, we first focus our attention on defining and studying both the local and global errors for the BEM. This is not a completely new study as the literature suggests, however, our approach is different. We use the basic foundations of the the method to define the errors. Since the method is a global method, first we use the interpolation error on each element to define what we have called a sublocal error. Then using the sublocal error we have defined the local error. Understanding the local errors enabled us study the global error. Theoretical and numerical results show that these errors are second order in grid size for both the constant and linear element cases. Then, having explored errors, we study a method for adaptive grid refinement for the BEM. Rather than using a truly nonuniform grid, we present a method called local defect correction (LDC) that is based on local uniform grid refinement. This method is already developed and documented for other numerical methods such as finite difference and finite volume methods but not for BEM. In the LDC method, the discretization on a composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain. At least one grid, the uniform global coarse grid, should cover the entire boundary. The size of the global coarse grid is chosen in agreement with the relatively smooth behaviour of the solution outside the hot spots. Then several uniform local fine grids each of which covers only a (small) part of the boundary are used in the hot spots. The grid sizes of the local grids are chosen in agreement with the behaviour of the continuous solution in that part of the boundary. The LDC method is an iterative process whereby a basic global discretisation is improved by local discretisations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continuous solution on the corresponding composite grid. We have shown how this discretisation can be achieved for the BEM. We apply the discretisation to an academic example to demonstrate its implementation and later show how to use it for an application as the ICCP system. The results show that it is a cheaper method than solving on a truly composite grid and converges in a single step.

An evaluation of in-silico methods for predicting solute partition in multiphase complex fluids – A case study of octanol/water partition coefficient

Seismic pressure point sources have been simulated near the surface of sand dunes in Oman. Real topographic elevation data was assembled from recorded field data in the region, in-filled with necessary interpolated data, to form the surface topography implemented. Recent results describing seismic sand curves for the region are employed in the near-surface medium definition, and a constant half space of realistic seismic velocities and densities are used for the simulated sub-sand medium. Topographic effects in the form of scattering and reflections from prominent sand dune topography are confirmed, showing differences between scattering from elongated and normally oriented sand dunes to the excited source orientation. 8th order Finite-Differences (F-Ds) are used to discretize full elastic wave equations in the velocity-stress formulation, reducing gradually the F-D order when approaching all numerical grid boundaries. At the free surface, I employed boundary conditions for arbitrary surface topography derived for the particle velocities, discretized by 2nd order F-Ds, and at the side and bottom boundaries exponential damping is applied for non-reflective boundaries. A rectangular grid is used to model the Cartesian wave equations from a curved grid by adding in the equations extra terms that depend on the arbitrary surface topography and its spatial derivatives. The curved grid is adapted to an arbitrary surface topography with gradually less curvature with depth towards the bottom of the grid, which is planar. The code is parallellized by domain decomposition through MPI (Message Passing Interface) parallellization between processors. This makes possible the time domain modelling of higher frequencies (even relevant for exploration) and larger areas than could be done otherwise. Only in the last 3-5 years has such complex modelling been commercially viable in hydrocarbon exploration.

In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases.

The paper introduces a local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. It is a truly meshless approach which uses the local sub clusters of domain nodes, named local domains of influence, for the calculation of the spatial partial derivatives. With the selected domain of influence, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. After the successful approximation function creation, all the needed differential operators can be constructed by applying an arbitrary operator on the approximation function. The main advantage of using the local method is that the overlapping domains of influence results in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node, instead of a large collocation matrix, and hence the sparse global derivative matrices from local contributions. So the method needs less computer storage and flops. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due to their suitability and better accuracy. To decrease accuracy degradations near boundaries, some different edge improvement techniques are also used. The method is tested on four benchmark problems on Burgers’ equation with mixed boundary conditions. Five nodded domains of influence are used in the local support. The numerical results for different value of Reynolds numbers (Re) is compared with analytical solution as well as some other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.