scholarly journals Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160809 ◽  
Author(s):  
A. A. Shah ◽  
W. W. Xing ◽  
V. Triantafyllidis
Keyword(s):  
Dimensional Space ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Problems ◽  
Dynamic Parameter ◽  
Partial Differential ◽  
Reduced Order ◽  
Linear And Nonlinear ◽  
Low Dimensional ◽  
Parameter Dependent ◽  
Parameter Values

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

scholarly journals Analytical Solutions for the Nonlinear Partial Differential Equations Using the Conformable Triple Laplace Transform Decomposition Method

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Shailesh A. Bhanotar ◽  
Mohammed K. A. Kaabar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Three Dimensions ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Linear And Nonlinear

In this paper, a novel analytical method for solving nonlinear partial differential equations is studied. This method is known as triple Laplace transform decomposition method. This method is generalized in the sense of conformable derivative. Important results and theorems concerning this method are discussed. A new algorithm is proposed to solve linear and nonlinear partial differential equations in three dimensions. Moreover, some examples are provided to verify the performance of the proposed algorithm. This method presents a wide applicability to solve nonlinear partial differential equations in the sense of conformable derivative.

Continuous Solution of Partial Differential Equations on Non-Rectangular and Non-Continuous Geometry

2014 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuous Solution ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Linear And Nonlinear

A high order continuous solution is obtained for partial differential equations on non-rectangular and non-continuous domain using Bézier functions. This is a mesh free alternative to finite element or finite difference methods that are normally used to solve such problems. The problem is handled without any transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error in the boundary conditions over the domain. The solution can be expressed in polynomial form. The effort is same for linear and nonlinear partial differential equations. The procedure is developed as a combination of symbolic and numeric calculation. The solution is obtained through the application of standard unconstrained optimization. A constrained approach is also developed for nonlinear partial differential equations. Examples include linear and nonlinear partial differential equations. The solution for linear partial differential equations is compared to finite element solutions from COMSOL.

Boundary tracing and boundary value problems: I. Theory

2007 ◽  
Vol 463 (2084) ◽  
pp. 1909-1924 ◽  
Author(s):  
Michael L Anderson ◽  
Andrew P Bassom ◽  
Neville Fowkes
Keyword(s):  
Differential Equation ◽  
Nonlinear Problems ◽  
Boundary Curve ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Boundary Constraints ◽  
Linear And Nonlinear ◽  
Boundary Tracing ◽  
Analytical Tools

Given an exact solution of a partial differential equation in two dimensions, which satisfies suitable conditions on the boundary of the domain of interest, it is possible to deform the boundary curve so that the conditions remain fulfilled. The curves obtained in this manner can be patched together in various ways to generate a remarkably broad range of domains for which the boundary constraints remain satisfied by the initial solution. This process is referred to as boundary tracing and works for both linear and nonlinear problems. This article presents a general theoretical framework for implementing the technique for two-dimensional, second-order, partial differential equations with a general flux condition imposed around the boundary. A couple of simple examples are presented that serve to demonstrate the analytical tools in action. Applications of more intrinsic interest are discussed in the following paper.

scholarly journals Optimal Combination of EEFs for the Model Reduction of Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jun Shuai ◽  
Xuli Han
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Model Reduction ◽  
Nonlinear Partial Differential Equations ◽  
Optimization Techniques ◽  
Optimal Combination ◽  
Nonlinear Pdes ◽  
Partial Differential ◽  
Tiny Amount ◽  
Low Dimensional

Proper orthogonal decomposition is a popular approach for determining the principal spatial structures from the measured data. Generally, model reduction using empirical eigenfunctions (EEFs) can generate a relatively low-dimensional model among all linear expansions. However, the neglectful modes representing only a tiny amount of energy will be crucial in the modeling for certain type of nonlinear partial differential equations (PDEs). In this paper, an optimal combination of EEFs is proposed for model reduction of nonlinear partial differential equations (PDEs), obtained by the basis function transformation from the initial EEFs. The transformation matrix is derived from straightforward optimization techniques. The present new EEFs can keep the dynamical information of neglectful modes and generate a lower-dimensional and more precise dynamical system for the PDEs. The numerical example shows its effectiveness and feasibility for model reduction of the nonlinear PDEs.

scholarly journals Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
Alexander Hay
Keyword(s):  
Finite Difference ◽  
Standard Method ◽  
Elliptic Cylinder ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Shape Sensitivity ◽  
Reduced Order ◽  
Flow Models ◽  
Low Dimensional ◽  
Parameter Values

Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

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