Two Dimensional and Time Varying Spatial Domains

Finite element and finite difference methods have been widely used, among other methods, to numerically solve the Fokker–Planck equation for investigating the time history of the probability density function of linear and nonlinear 2d and 3d problems; also the application to 4d problems has been addressed. However, due to the enormous increase in computational costs, different strategies are required for efficient application to problems of dimension ≥3. Recently, a stabilized multi-scale finite element method has been effectively applied to the Fokker–Planck equation. Also, the alternating directions implicit method shows good performance in terms of efficiency and accuracy. In this paper various finite difference and finite element methods are discussed, and the results are compared using various numerical examples.

PDE backstepping control of one-dimensional heat equation with time-varying domain

A 2D algebraic grid generation technique is presented for generating grid points for time-varying and irregular spatial domains. The method is based on the “four-boundary technique” of Vinokur and Lombard (1983) and is an extension of the “two-boundary technique for time-varying spatial domains” method developed by Yang and Shih (1986). Since it is an algebraic approach, the method is very efficient and can easily be extended to time-varying 3D spatial domains. To show the feasibility and the usefulness of the grid generation technique, (1) it is applied to the problem of generating grids for the combustion chamber above an internal combustion engine piston with bowl-in-cavity, and (2) numerical solutions are obtained for the fluid motion in the combustion chamber for intake and compression strokes.

Conventionally, the problem of studying the transport of water, heat, and solute in soil or groundwater systems has been numerically solved using finite difference (FD) or finite element (FE) methods. FE methods are attractive over FD methods because they are geometrically flexible. However, recent studies demonstrate that spectral collocation (SC) methods converge exponentially faster than FD or FE methods using a few grid points or on coarse grids. This work proposes and applies a multivariate spectral local quasilinearization method (MV-SLQLM) to model the transportation and interaction of soil moisture, heat, and solute concentration in a nonbare soil ridge. The MV-SLQLM uses a quasilinearization method (QLM) to linearize any nonlinear equations and then employs a local linearization method (LLM) to decouple the linearized system of PDEs to form a sequence of equations that are solved in a computationally efficient manner. The MV-SLQLM is thus an extension of the bivariate spectral local linearization method (BI-SLLM) that fails to deal with a 2D problem and is a modification of the MV-SQLM whose efficiency is compromised when operating on high dense solution matrices. We use the residual error norms of the difference between successive iterations to affirm convergence to the expected solution. To illustrate the application and check the solution accuracy, we conduct systematic analyses of the effect of model parameters on distribution profiles. Findings are in good agreement with theory and literature, thereby revealing suitability of the MV-SLQLM to solve coupled nonlinear PDEs with environmental fluid dynamics applications.

In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pade approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.

This paper deals with the multi-scale optimal control of transport-reaction systems with the underlying dynamics governed by the second order rigid body dynamics, coupled with the parabolic partial differential equations (PDEs) with time-varying spatial domains, developed by considering the first principles dynamical equations for continuum mechanics. A functional theory is employed to explore the process model time-varying features, which lead to the characterization of the time varying spatial operator as a Riesz-spectral operator. This characterization facilitates the formulation of the optimal control problem where the infinite-dimensional system associated with the time-varying spatial operator is coupled with a finite-dimensional system describing the motion of the domain. The temperature control of the underlying transportreaction dynamics is realized through the optimal control law regulating the trajectory of the domain boundary coupled with the optimal heating input applied along the domain. The optimal control law associated with the domain's boundary is obtained as a solution to the algebraic Riccati equation, while the optimal control law associated with the temperature regulation is obtained as a solution of a time-dependent Ricatti equation.

In this work a PDE backstepping-based control law for one-dimensional unstable heat equation with time-varying spatial domain is developed. The underlying parabolic partial differential equation (PDE) with time-varying domain is the model emerging from process control applications such as crystal growth. In backstepping control law synthesis, a characteristic feature is that the PDE describing the transformation kernel of the associated Volterra integral is time-dependent. In this work, the kernel PDE is solved numerically and the state-feedback controller is simulated for the application of temperature regulation in the Czochralski crystal growth process.

Accurate and efficient numerical wave propagation is important in many areas of study such as computational aero-acoustics (CAA). While dissipation and dispersion errors influence the accuracy of a method, efficiency can be assessed by convergence rates and effective adaptability to different mesh structures. Finite difference and finite element methods are commonly used numerical schemes in CAA. Finite difference methods have the advantages of ease of use as well as high order convergence, but often require a uniform grid, and stable boundary closure can be non-trivial. Finite element methods adapt well to different mesh structures but can become difficult to implement as the order of approximation increases. In this research we formulate a numerical method that has high-order convergence, with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the Discontinuous Galerkin Method (DGM) applied to the hyperbolic equation. Finite difference type schemes applicable to non-uniform grids are proposed. The schemes will be referred to as DGM-FD schemes. These schemes inherit, naturally, some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Two grid structures are studied. In the first structure, a regular, but non-uniform, finite difference type grid is assumed. In the second structure, some grid points are double-valued and the derivative scheme has a shortened stencil. Fourth-order upwind and third order central schemes are presented as examples of the first grid structure. Fifth-order upwind schemes are derived for the second structure. For non-linear equations, flux finite difference formula are given where no explicit upwind and downwind split of the flux is needed. This is in contrast to existing upwind finite difference schemes in the literature. Stability of the schemes with boundary closures and the super-accuracy for wave propagation problems are investigated and validated. The new schemes are demonstrated by numerical examples including the linearized acoustic waves, the solution of non-linear Burger's equation and the flat-plate boundary layer problem.

Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity

This work considers the modelling problem of the dynamics of overhead cranes with flexible cable and load hoisting or lowering during crane travel. The analysis includes the transverse vibrations of the flexible cable and trolley motion. A set of nonlinear ordinary differential equations governing the motion of the crane system with time-varying spatial domain is derived via the calculus of variation and Hamilton’s principle. A variable-domain finite element method is used to discretize the nonlinear system. A proportional -derivative controller is suggested to drive the crane to a desired destination. Numerical simulations are provided to show the effectiveness of the developed model and to illustrate the results.

The scope of this book is to present well known simple and advanced numerical methods for solving partial differential equations (PDEs) and how to implement these methods using the programming environment of the software package Diffpack. A basic background in PDEs and numerical methods is required by the potential reader. Further, a basic knowledge of the finite element method and its implementation in one and two space dimensions is required. The authors claim that no prior knowledge of the package Diffpack is required, which is true, but the reader should be at least familiar with an object oriented programming language like C++ in order to better comprehend the programming environment of Diffpack. Certainly, a prior knowledge or usage of Diffpack would be a great advantage to the reader.

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

The analysis of adhesively bonded joints started in 1938 with the closed-form model of Volkersen. The equilibrium equation of a single lap joint led to a simple governing differential equation with a simple algebraic equation. However, if there is yielding of the adhesive and/or the adherends and substantial peeling is present, a more complex model is necessary. The more complete is an analysis, the more complicated it becomes and the more difficult it is to obtain a simple and effective solution. The finite element (FE) method, the boundary element (BE) method and the finite difference (FD) method are the three major numerical methods for solving differential equations in science and engineering. These methods have also been applied to adhesive joints, especially the FE method. This book deals with the most recent numerical modelling of adhesive joints. Advances in damage mechanics and extended finite element method are described in the context of the FE method with examples of application. The classical continuum mechanics and fracture mechanics approach are also introduced. The BE method and the FD method are also discussed with indication of the cases they are most adapted to. There is not at the moment a numerical technique that can solve any problem and the analyst needs to be aware of the limitations involved in each case.

The development of groundwater models solving actual engineering problems is of great interest as well with regard to physical as with regard to economical aspects. These models enable a more precise treatment of problems during the stage of planning. Furthermore, these models assure a permanent control of finished steps. Generally, the flow through porous media is a three-dimensional process depending on various parameters. Normally, this process is so complicated that it is not possible to describe the situation in nature in a general form. Therefore, it is necessary to develop some simplified physical models of groundwater flow which are adapted to relevant individual circumstances. These refer for example to steady or unsteady flow, to flow with or without a free surface, to flow in an isotropic or anisotropic aquifer, to one-, two- or three-dimensional flow. The solution of these different physical models can be obtained by use of analytical, analog or digital methods, which consider the typical physical conditions. Such a concept of a physical model and a corresponding solution method is to be defined as a socalled analytical, analog or digital flow model through porous media. As the organization of flow models through porous media causes generally high costs, which c1epend mainly on the choice of the solution method, it is of great interest, which of the different solution methods is suitable for a given problem. It was possible to prove by a cost-effectiveness-analysis, that digital solution methods generally are considerably more effective than analytical and analogous methods. As digital solution methods, one can use either the method of finite elements or the method of finite differences. Both methods produce discrete solutions of given problems. The comparison of both methods is to be done by significant criteria. These are: 1. the required core storage; 2. the required computing time; 3. the flexibility of the methods approximating the problems. As the core storage and the computing time c1irectly depend on the organization technique and the algorithm solving linear equation systems, special organization techniques are to be discussed. The flexibility is to be seen in clependence on physical problems. The comparison of both methods (methods of finite elements and finite differences, respectively) shows, that the method of finite differences is much more better with regard to organization and programming aspects, especially however, with regard to economical aspects (Jess required core storage, less computing time). In general, flow problems through porous media should be solved by the method of finite differences. For all problems, however, which involve an automatic search of free surfaces, the method of finite elements appears to be more suitable, because only its organization can realize these problems.