Nanoemulsion drug delivery systems are advanced modes for delivering and improving the bioavailability of hydrophobic drugs and the drug which have high first pass metabolism. The nanoemulsion can be ...prepared by both high energy and low energy methods. High energy method includes high-pressure homogenization, microfluidization, and ultrasonication whereas low energy methods include the phase inversion emulsification method and the self-nanoemulsification method. Low energy methods should be preferred over high energy methods as these methods require less energy, so are more efficient and do not require any sophisticated instruments. However high energy methods are more favorable for food grade emulsion as they require lower quantities of surfactant than low energy methods. Techniques for formulation of nanoemulsion drug delivery system are overlapping in nature, especially in the case of low energy methods. In this review, we have classified different methods for formulation of nanoemulsion systems based on energy requirements, nature of phase inversion, and self-emulsification.
•Using machine learning to solve PDEs has seen increased interest in solid mechanics.•Current methods fail to reliably resolve concentration features in hyperelasticity.•Mixed Deep Energy Method ...(mDEM) is introduced that is able to resolve concentrations.•mDEM utilizes stress as an additional output to displacements.•Method requires the approximation of lower order derivatives compared to PINN.
The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations, i.e. domains with holes, notches, etc. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads, but additionally offers unique capabilities for the solution of inverse problems and parameter estimation in the context of hyperelasticity.
In this paper, we present a deep autoencoder based energy method (DAEM) for the bending, vibration and buckling analysis of Kirchhoff plates. The DAEM exploits the higher order continuity of the DAEM ...and integrates a deep autoencoder and the minimum total potential principle in one framework yielding an unsupervised feature learning method. The DAEM is a specific type of feedforward deep neural network (DNN) and can also serve as function approximator. With robust feature extraction capacity, the DAEM can more efficiently identify patterns behind the whole energy system, such as the field variables, natural frequency and critical buckling load factor studied in this paper. The objective function is to minimize the total potential energy. The DAEM performs unsupervised learning based on generated collocation points inside the physical domain so that the total potential energy is minimized at all points. For the vibration and buckling analysis, the loss function is constructed based on Rayleigh’s principle and the fundamental frequency and the critical buckling load is extracted. A scaled hyperbolic tangent activation function for the underlying mechanical model is presented which meets the continuity requirement and alleviates the gradient vanishing/explosive problems under bending. The DAEM is implemented using Pytorch and the LBFGS optimizer. To further improve the computational efficiency and enhance the generality of this machine learning method, we employ transfer learning. A comprehensive study of the DAEM configuration is performed for several numerical examples with various geometries, load conditions, and boundary conditions.
•Deep autoencoder based energy method (DAEM) with tailored activation function.•Stable and accurate results without gradient vanishing/exploding problems.•Unsupervised DAEM applied to Kirchhoff plates.
In this work, we present a Parametric Deep Energy Method (P-DEM) for elasticity problems accounting for strain gradient effects. The approach is based on physics-informed neural networks (PINNs) for ...the solution of the underlying potential energy. Therefore, a cost function related to the potential energy is subsequently minimized. P-DEM does not need any classical discretization and requires only a definition of the potential energy, which simplifies the implementation. Instead of training the model in the physical space, we define a parametric/reference space similar to isoparametric finite elements, which is in our example a unit square. The inputs are naturally normalized preventing the vanishing gradient problem and leading to much faster convergence compared to the original DEM. Forward–backward mapping is established by means of NURBS basis functions. Another advantage of this approach is that Gauss quadrature can be employed to approximate the total potential energy, which is the loss function calculated in the parametric domain. Backpropagation available in PyTorch with automatic differentiation is performed to calculate the gradients of the loss function with respect to the weights and biases. Once the network is trained, a numerical solution can be obtained in the reference domain and then is mapped back to the physical domain. The performance of the method is demonstrated through various numerical benchmark problems in elasticity and compared to analytical solutions. We also consider strain gradient elasticity, which poses challenges to conventional finite elements due to the requirement for C1 continuity.
•An extension of the deep energy approach has been applied to elasticity accounting for strain gradient effects with the aid of neural networks•Physical domains are mapped to the reference domain for better training performance•Forward and backward mapping are achieved through NURBS basis functions•The approximate loss energy is conveniently calculated utilizing Gauss quadrature.
This paper continues a series of studies providing stability crite¬ria for quasigeostrophic forced zonal flows in in the presence of lateral diffusion and bottom dissipation of the vertical ...vorticity. We study the Lyapunov stability of a stationary and longitude independent ba¬sic flow, obtaining linear and nonlinear stability criteria expressed in terms of the maximum shear of the basic flow and/or its meridional derivative, extending some previous results.
•An analytical model of a 3D re-entrant auxetic cellular structure was established.•Overlapping of struts and axial extension/compression was taken into consideration.•Strut slenderness significantly ...influence on deformation mechanisms of structures.•Axial extension/compression play decisive role on the lateral Poisson's ratio.
In this work, an analytical model of a 3D re-entrant auxetic cellular structure has been established based on energy method. In the model the overlapping of the struts as well as axial extension or compression (mostly neglected in former studies) were taken into consideration to make the model applicable when the struts are relative stubby which is common in engineering designs. Analytical solutions for the modulus and Poisson's ratios of the cellular structure in all principal directions were deduced. To validate the analytical model in present study, numerical calculations using brick elements were performed on unit cell models with periodic boundary conditions, and comparisons of the present model with analytical formulae and experimental results available in former literatures were also conducted. The results show that when the struts are slender enough, the bending of the struts play decisive role on the deformation of the structure and other mechanisms can be ignored; while when the struts become relative stubby, all the mechanisms including bending, shearing and axial loading need to be considered; The often-ignored axial extension or compression term may even play decisive role on determining the lateral Poisson's ratio of the structure when the struts are relative stubby.
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In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the
L1 discretization is applied for the ...time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method.
A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of ...the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost
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This paper establishes a geometrically nonlinear bending analysis framework using the deep energy method and the classical laminated plate theory (CLPT) for laminated plates. Inspired by the transfer ...learning technique, a load applied to a laminated plate can be divided into multiple load steps. The network parameters for the current load step, with the exception of the initial step, are initialized by inheriting values from their preceding steps. Including both von Kármán and Green-Lagrange strains, the plate strains are computed using the automatic differentiation and integrated along the thickness direction per laminate plate based on the constitutive theory. By combining the outputs of neural network, the external potential energy can be obtained, and the optimized network parameters are given by minimizing the total system potential energy of the laminated plate. In order to validate the proposed approach, several numerical examples are calculated, and the present solutions are compared with those given by the literature and the Finite Element Analysis (FEA). The results show that the proposed approach is indeed feasible, can reach high levels of precision under varying loads while offering a simplified calculation strategy.
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