Beam analogy for optimal control of linear dynamic systems

 Optimal control problems for linear dynamic systems with quadratic performance index are solved using the beam analogy. The governing equations for the optimal maneuver are derived in the form of coupled fourth order differential equations in the time domain. These equations are uncoupled using modal variables. Next, each independent equation is made analogous to the corresponding problem of a beam on an elastic foundation. The beam problem in the spatial domain is solved using standard FEM software. Finally the FEM results are transferred back to the time domain where they represent the optimal trajectories and controls for the dynamic system. Received 12 October 1999     

Solving boundary-value problems using hp-version finite elements in time

An hp-version finite element method for one-dimensional boundary value problems is presented. The method is based on a similar approach developed by the authors for solution of optimal control problems. The primary applications for the methodology include two-point- and multi-point-boundary-value problems, for example, in the time domain. Results presented for a 7-state/3-phase missile problem show that the method is very efficient for time-marching applications. Furthermore, it easily solves time-domain problems with discontinuities in the system equations and/or in the states, where the time at which these jumps (i.e. ‘events’) take place is determined by equations that govern the states. An example involving friction with intermittent sticking is presented to illustrate the power of the method. © 1998 John Wiley & Sons, Ltd.     

Suppression of nonlinear composite panel flutter with active/passive hybrid piezoelectric networks using finite element method

For the suppression of nonlinear panel flutter, a new optimal active/passive hybrid control design with piezoceramic actuators is proposed using finite element methods. This approach has the advantages of both active (high performance, feedback action) and passive (stable, low power requirement) systems. Piezoceramic actuators are connected in series with an external voltage source and a passive resonant shunt circuit which consists of an inductor and resistor. The shunt circuit should be tuned correctly to suppress the flutter effectively with less control effort as compared to purely active control. To obtain the best effectiveness, active control gains are simultaneously optimized together with the value of the resistor and inductor through a sequential quadratic programming method. The governing equations of the electromechanically coupled composite panel flutter are derived through an extended Hamilton’s principle, and a finite element discretization is carried out. The adopted aerodynamic theory is based on the quasi-steady first-order piston theory, and the von Kármán nonlinear strain–displacement relation is used. Nonlinear modal equations are obtained through a modal reduction technique. Optimal control design is based on linear modal equations of motion, and numerical simulations are based on nonlinear-coupled modal equations. Using the Newmark integration method, suppression results of a hybrid control and a purely active control are presented in the time domain.     

Superconvergence of Fully Discrete Finite Elements for Parabolic Control Problems with Integral Constraints

A quadratic optimal control problem governed by parabolic equations with integral constraints is considered. A fully discrete finite element scheme is constructed for the optimal control problem, with finite elements for the spatial but the backward Euler method for the time discretisation. Some superconvergence results of the control, the state and the adjoint state are proved. Some numerical examples are performed to confirm theoretical results.     

Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

复合材料热传导系数均匀化计算的实现方法

均匀化理论可以有效预测周期性结构复合材料的等效热传导系数,然而其控制方程的载荷项形式特殊,通用有限元软件中没有对应的载荷形式,难以直接求解.提出一种本构关系及场变量的类比方法,证明了在此类比下等效热传导系数均匀化方程与等效弹性模量均匀化方程是等价的.根据求解等效弹性模量均匀化方程的热应变法,提出一种新的等效热传导系数均匀化方程数值求解方法.以ABAQUS为平台,预测单向纤维复合材料以及金属蜂窝夹芯板的等效热传导系数,计算结果与参考值吻合良好.该方法为基于通用有限元软件的复合材料等效热传导系数的均匀化计算提供了简便途径.     

Finite element/finite volume approaches with adaptive time stepping strategies for transient thermal problems

Transient thermal analysis of engineering materials and structures by space discretization techniques such as the finite element method (FEM) or finite volume method (FVM) lead to a system of parabolic ordinary differential equations in time. These semidiscrete equations are traditionally solved using the generalized trapezoidal family of time integration algorithms which uses a constant single time step. This single time step is normally selected based on the stability and accuracy criteria of the time integration method employed. For long duration transient analysis and/or when severe time step restrictions as in nonlinear problems prohibit the use of taking a larger time step, a single time stepping strategy for the thermal analysis may not be optimal during the entire temporal analysis. As a consequence, an adaptive time stepping strategy which computes the time step based on the local truncation error with a good global error control may be used to obtain optimal time steps for use during the entire analysis. Such an adaptive time stepping approach is described here. Also proposed is an approach for employing combinedFEM/FVM mesh partitionings to achieve numerically improved physical representations. Adaptive time stepping is employed thoughout to practical linear/nonlinear transient engineering problems for studying their effectiveness in finite element and finite volume thermal analysis simulations. Additional support and computing times were furnished by Minnesota Supercomputer Institute at the University of Minnesota.     

求解粘弹性问题的时域自适应等几何比例边界有限元法

提出一种基于分段时域自适应算法和等几何分析的求解粘弹性问题的数值方法。利用时域分段展开,建立了递推格式的比例边界元求解方程,环向比例边界采用等几何技术离散,在继承常规比例边界有限元半解析、便于处理应力奇异性/无限域问题等优点的同时,可更准确地描述几何边界,由此进一步提高了计算精度;在时域,通过分段时域自适应计算,保证不同时间步长下的计算精度。通过数值算例,从计算精度、收敛性等方面,对所提方法的有效性进行了验证。     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.     

光滑有限元的声学研究：时域和频域分析

摘　要：在使用有限元进行声场的数值模拟中,存在着两个主要误差,一个是数值方法中常规的插值误差,另外一个是计算声学中所特有的耗散误差(dispersion error),后者则是影响声学模拟仿真置信度的最重要因素。产生耗散误差的本质原因是由于有限元的数值模型刚度“偏硬”造成的。为了控制耗散误差,最重要的是使数值模型更好的反映真实模型。本文采用了一种基于边光滑的有限元方法(ES-FEM)来对声场的时域和频域进行数值模拟研究。该方法只采用对复杂问题域适应性很强的三角形网格,通过引进基于边的广义梯度光滑技术,能够使得有限元系统得到适当的“软化”。关于时域和频域的算例表明了在使用同样网格的情况下,本方法在声学模拟中的精度都要比有限元模型的高。     

A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.     

Solving viscoelastic problems by combining SBFEM and a temporally piecewise adaptive algorithm

A combination of the scaled boundary finite element method (SBFEM) with a temporally piecewise adaptive algorithm is exploited to solve viscoelastic problems. By expanding variables at a discretized time interval, a coupled spatial–temporal problem is decoupled into a series of recursive spatial problems, which are solved by SBFEM, and a piecewise adaptive process in the time domain is realized via the change of expansion powers. Numerical verification, including the cases involving stress singularity, infinite domain, and inhomogeneous medium, are provided in comparison with analytical or ABAQUS-based solutions.     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical approximation of optimal control for distributed diffusion Hopfield neural networks

This work presents a numerical approximation of optimal control problems for non‐linear distributed Hopfield Neural Network equations with diffusion term. For one spatial dimensional case, a semi‐discrete numerical algorithm was constructed to find optimal control variable using finite element discretization, updated conjecture gradient iteration method. Furthermore, experiments demonstration will be implemented to show the effectiveness and stability through 3D graphics simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

High-order cut discontinuous Galerkin methods with local time stepping for acoustics

We propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous Galerkin method for spatial discretization and the arbitrary derivative method with local time stepping (LTS) for time integration. The method is based on a cut finite element approach of high order and uses level set functions to describe curved immersed interfaces. We study under which conditions and to what extent small time step sizes balance cut instabilities, which are present especially for high-order spatial discretizations. This is done by analyzing eigenvalues and critical time steps for representative cuts. If small time steps cannot prevent cut instabilities, stabilization by means of cell agglomeration is applied and its effects are analyzed in combination with local time step sizes. Based on two examples with general cuts, performance gains of the LTS over the global time stepping are evaluated. We find that LTS combined with cell agglomeration is most robust and efficient.     

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

A methodology is introduced for rapid reduced‐order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy. Copyright © 2005 John Wiley & Sons, Ltd.     

Investigation on a Two-dimensional Generalized Thermal Shock Problem with Temperature-dependent Properties

The dynamic response of a two-dimensional generalized thermoelastic problem with temperature-dependent properties is investigated in the context of generalized thermoelasticity proposed by Lord and Shulman. The governing equations are formulated, and due to the nonlinearity and complexity of the governing equations resulted from the temperature-dependent properties, a numerical method, i.e., finite element method is adopted to solve such problem. By means of virtual displacement principle, the nonlinear finite element equations are derived. To demonstrate the solution process, a thermoelastic half-space subjected to a thermal shock on its bounding surface is considered in detail. The nonlinear finite element equations for this problem are solved directly in time domain. The variations of the considered variables are obtained and illustrated graphically. The results show that the effect of the temperature-dependent properties on the considered variables is to reduce their magnitudes, and taking the temperature-dependence of material properties into consideration in the investigation of generalized thermoelastic problem has practical meaning in predicting the thermoelastic behaviors accurately. It can also be deduced that directly solving the nonlinear finite element equations in time domain is a powerful method to deal with the thermoelastic problems with temperature-dependent properties.     

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Topology optimization by a time‐dependent diffusion equation

Most topology optimization problems are formulated as constrained optimization problems; thus, mathematical programming has been the mainstream. On the other hand, solving topology optimization problems using time evolution equations, seen in the level set‐based and the phase field‐based methods, is yet another approach. One issue is the treatment of multiple constraints, which is difficult to incorporate within time evolution equations. Another issue is the extra re‐initialization steps that interrupt the time integration from time to time. This paper proposes a way to describe, using a Heaviside projection‐based representation, a time‐dependent diffusion equation that addresses these two issues. The constraints are treated using a modified augmented Lagrangian approach in which the Lagrange multipliers are updated by simple ordinary differential equations. The proposed method is easy to implement using a high‐level finite element code. Also, it is very practical in the sense that one can fully utilize the existing framework of the code: GUI, parallelized solvers, animations, data imports/exports, and so on. The effectiveness of the proposed method is demonstrated through numerical examples in both the planar and spatial cases. Copyright © 2012 John Wiley & Sons, Ltd.