Model Order Reduction for a Piecewise Linear System Based on Dynamic Mode Decomposition

This paper presents a data-driven model order reduction strategy for nonlinear systems based on dynamic mode decomposition (DMD). First, the theory of DMD is briefly reviewed and its extension to model order reduction of nonlinear systems based on Galerkin projection is introduced. The proposed method utilizes impulse response of the nonlinear system to obtain snapshots of the state variables, and extracts dynamic modes that are then used for the projection basis vectors. The equations of motion of the system can then be projected onto the subspace spanned by the basis vectors, which produces the projected governing equations with much smaller number of degrees of freedom (DOFs). The method is applied to the construction of the reduced order model (ROM) of a finite element model (FEM) of a cantilevered beam subjected to a piecewise-linear boundary condition. First, impulse response analysis of the beam is conducted to obtain the snapshot matrix of the nodal displacements. The DMD is then applied to extract the DMD modes and eigenvalues. The extracted DMD mode shapes can be used to form a reduction basis for the Galerkin projection of the equation of motion. The obtained ROM has been used to conduct the forced response calculation of the beam subjected to the piecewise linear boundary condition. The results obtained by the ROM agree well with that obtained by the full-order FEM model.

DispHred: A Server to Predict pH-Dependent Order–Disorder Transitions in Intrinsically Disordered Proteins

The natively unfolded nature of intrinsically disordered proteins (IDPs) relies on several physicochemical principles, of which the balance between a low sequence hydrophobicity and a high net charge appears to be critical. Under this premise, it is well-known that disordered proteins populate a defined region of the charge–hydropathy (C–H) space and that a linear boundary condition is sufficient to distinguish between folded and disordered proteins, an approach widely applied for the prediction of protein disorder. Nevertheless, it is evident that the C–H relation of a protein is not unalterable but can be modulated by factors extrinsic to its sequence. Here, we applied a C–H-based analysis to develop a computational approach that evaluates sequence disorder as a function of pH, assuming that both protein net charge and hydrophobicity are dependent on pH solution. On that basis, we developed DispHred, the first pH-dependent predictor of protein disorder. Despite its simplicity, DispHred displays very high accuracy in identifying pH-induced order/disorder protein transitions. DispHred might be useful for diverse applications, from the analysis of conditionally disordered segments to the synthetic design of disorder tags for biotechnological applications. Importantly, since many disorder predictors use hydrophobicity as an input, the here developed framework can be implemented in other state-of-the-art algorithms.

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

Collocation Trefftz Method for the Heat Conduction Issue in Irregular Domain with Non-Linear Boundary Condition

Rules for nonlinear borders of irregular domain is the thorny issues when using analytical method for solving mathematical and physical equations. On the basis of solution in the form of separated variables, the border of arbitrary shape with non-orthogonal boundary will be separated into a limited number of discrete points, and then direct assignment for the form solution at each of the discrete points on the border according to boundary conditions, at every discrete points on the border can establish an equation. If the number of discrete points on the border is equal with truncated series after retained series, all coefficients of the form solution can be determined and the problem solved. This paper use Laplace equation as an example to illustrate Collocation Trefftz Method can solve certain steady heat conduction problems within irregular domain with non-linear border.

A Three-Dimensional Finite Element Method for Nonsteady State Temperature Distribution in the Rolling Process

The temperature is a key factor that affects the metal deformation and the material property in the rolling process. The metal deformation is often carried along with the variety of temperature. Moreover, the plastic work is converted into the thermal energy during the process of the metal deformation. Therefore, the numerical simulation of the rolling process should take the temperature factors into consideration to improve the prediction accuracy. In this paper, we use the full three-dimensional Rigid-plastic finite element method to predict the temperature distribution which relate to linear or non-linear boundary condition of the free surface. Based on the simulation, the impact of radiant heat-transfer coefficient on the temperature prediction can be obtained. The comparison of the calculated results between the linear and non-linear boundary conditions demonstrates that the temperature obtained on the linear boundary condition has higher accurateness than that obtained on the non-linear boundary condition.