Scattered data approximation by regular grid weighted smoothing

Scattered data approximation refers to the computation of a multi-dimensional function from measurements obtained from scattered spatial locations. For this problem, the class of methods that adopt a roughness minimization are the best performing ones. These methods are called variational methods and they are capable of handling contrasting levels of sample density. These methods express the required solution as a continuous model containing a weighted sum of thin-plate spline or radial basis functions with centres aligned to the measurement locations, and the weights are specified by a linear system of equations. The main hurdle in this type of method is that the linear system is ill-conditioned. Further, getting the weights that are parameters of the continuous model representing the solution is only a part of the effort. Getting a regular grid image requires re-sampling of the continuous model, which is typically expensive. We develop a computationally efficient and numerically stable method based on roughness minimization. The method leads to an algorithm that uses standard regular grid array operations only, which makes it attractive for parallelization. We demonstrate experimentally that we get these computational advantages only with a little compromise in performance when compared with thin-plate spline methods.     

A mixed least squares method for solving problems in linear elasticity: formulation and initial results

In this paper a mixed least squares finite element method for solving problems in linear elasticity is proposed. The developed numerical technique allows the use of separate unknowns for displacements and stresses, discontinuous interpolation functions for displacements, and the resulting linear system has a symmetric and positive definite coefficient matrix. The approximate solution of the linear elasticity problem is obtained by minimization of a least squares functional based on the constitutive equations and equations of equilibrium. The proposed method is implemented in an original computer code written in C programming language. Its performance is tested on classical examples from theory of elasticity with well-known exact analytical solutions. Results from the implementation of a constant displacement-bilinear stress element and bilinear displacement-bilinear stress element are discussed.     

Non-linear structural analysis by dynamic relaxation

Dynamic relaxation, an iterative method for use with digital computers, is described and is shown to be suitable for the solution of a system of linear equations and in particular for such problems derived from structural frame analysis. It is further shown that the method may be modified to include non-linear equations relating to these problems. Some specific examples of linear and non-linear solutions are given and comparisons are made with another computer method which performs the same tasks.     

Iterative procedures for improving penalty function solutions of algebraic systems

The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large-scale finite element codes are offered.     

Formulation and solution of hierarchical finite element equations

The paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems. Static and eigenvalue linear elastic problems as well as convection–diffusion problems are studied. The hierarchical formulation is well suited for adaptive procedures. For the convection-diffusion problem the hierarchical approximation is made in time only. Different hierarchical functions are proposed for different types of problems. Both weighted residual and least-squares formulations are applied. A combination of these two gives a penalty method with a constraint equation corresponding to the least-squares method. A whole class of time integration formulae is obtained. These are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain. If a linear discontinuous hierarchical base function is used in the Galerkin weak formulation, the method so obtained corresponds to the discontinuous Galerkin method in time and is especially suited for convection dominated problems. The streamline-diffusion method is found to be the aforementioned penalty method. This paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation. Properties of these systems arising from static problems are investigated. The paper presents some new possibilities for iterative solution of hierarchic element equations, and different procedures are compared in a numerical example. Finally, a simple ID convection-diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow.     

The simple and multiple job assignment problems

This paper addresses two real-life assignment problems. In both cases, the number of employees to whom tasks should be assigned is significantly greater than the number of tasks. In the simple job assignment problem, at most one task (job) should be assigned to each employee; this constraint is relaxed in the multiple job assignment problem. In both cases, the goal is to minimize the time the last task is completed: these problems are known as Bottleneck Assignment Problems (BAPs for short). We show that the simple job assignment problem can be solved optimally using an iterative approach based on dichotomy. At each iteration, a linear programming problem is solved: in this case the solution is integer. We propose a fast heuristic to solve the multiple job assignment problem, as well as a branchand-bound approach which leads to an optimal solution. Numerical examples are presented. They show that the heuristic is satisfactory for the application at hand.     

Control of the freezing interface motion in two-dimensional solidification processes using the adjoint method

The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two-dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved. The so-called two-dimensional inverse Stefan design problem is formulated as an infinite-dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two-dimensional numerical examples are analysed to demonstrate the performance of the present method.     

Non‐convex dual forms based on exponential intervening variables,with application to weight minimization

We study the weight minimization problem in a dual setting. We propose new dual formulations for non‐linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expansions in terms of exponential intervening variables. These, generally, nonconvex approximations are formulated in terms of intervening variables with negative exponents, and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization (SAO) framework. Problems in structural optimization are traditionally solved using SAO algorithms, like the method of moving asymptotes, which require the approximate subproblems to be strictly convex. Hence, during solution, the nonconvex problems are approximated using convex functions, and this process may in general be inefficient. We argue, based on Falk's definition of the dual, that it is possible to base the dual formulation on nonconvex approximations. To this end we reintroduce a nonconvex approach to the weight minimization problem originally due to Fleury, and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables. We show in each case that the dual is well defined for the form concerned, which may consequently be of use to the future code developers. Copyright © 2009 John Wiley & Sons, Ltd.     

A statistical view of iterative methods for linear inverse problems

In this article, we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods, from a statistical point of view. The basic purpose of the paper is to develop adaptive model selection techniques for determining the regularization parameters, i.e., the iteration index. We assume observations are taken over a fixed grid and we consider solutions over a sequence of finite-dimensional subspaces. Based on concentration inequalities techniques, we derive non-asymptotic optimal upper bounds for the mean square error of the proposed estimator.     

Multidisciplinary design optimization based on independent subspaces

Optimization has been successfully applied to systems with a single discipline. Since many disciplines are involved in a coupled fashion in modern engineering, multidisciplinary design optimization (MDO) technology has been developed. MDO algorithms are designed to solve the coupled aspects generated from the interdisciplinary relationship. In a general MDO algorithm, a large design problem is decomposed into smaller ones which can be easily solved. Although various methods have been proposed for MDO, research is still in the early stage. This study proposes a new MDO method which is named MDO based on independent subspaces (MDOIS). Many real engineering problems consist of physically separate components and they can be independently designed. The inter‐relationship occurs through coupled physics. MDOIS is developed for such problems. In MDOIS, a large system is decomposed into small subsystems. The coupled aspects are solved via system analysis which solves the coupled physics. The algorithm is mathematically validated by showing that the solution satisfies the Karush–Kuhn–Tucker condition. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

Efficient numerical methods for the large‐scale,parallel solution of elastoplastic contact problems

Quasi‐static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid‐preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three‐dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns. Copyright © 2015 John Wiley & Sons, Ltd.     

A boundary knot method for harmonic elastic and viscoelastic problems using single-domain approach

The boundary knot method is a promising meshfree, integration-free, boundary-type technique for the solution of partial differential equations. It looks for an approximation of the solution in the linear span of a set of specialized radial basis functions that satisfy the governing equation of the problem. The boundary conditions are taken into account through the collocation technique. The specialized radial basis function for harmonic elastic and viscoelastic problems is derived, and a boundary knot method for the solution of these problems is proposed. The completeness issue regarding the proposed set of radial basis functions is discussed, and a formal proof of incompleteness for the circular ring problem is presented. In order to address the numerical performance of the proposed method, some numerical examples considering simple and complex domains are solved.     

Adaptive multivlevel methods in three space dimensions

We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.     

A mixed least squares method for solving problems in linear elasticity: theoretical study

 In a previous paper we proposed a mixed least squares method for solving problems in linear elasticity. The solution to the equations of linear elasticity was obtained via minimization of a least squares functional depending on displacements and stresses. The performance of the method was tested numerically for low order elements for classical examples with well known analytical solutions. In this paper we derive a condition for the existence and uniqueness of the solution of the discrete problem for both compressible and incompressible cases, and verify the uniqueness of the solution analytically for two low order piece-wise polynomial FEM spaces. Received: 20 January 2001 / Accepted: 14 June 2002 The authors gratefully acknowledge the financial support provided by NASA George C. Marshall Space Flight Centre under contract number NAS8-38779.     

Hierarchical model reduction at multiple scales

A hierarchical model reduction approach aimed at reducing computational complexity of non‐linear homogenization at multiple scales is developed. The method consists of the following salient features: (1) formulation of non‐linear unit cell problems at multiple scales in terms of eigendeformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized non‐linear equilibrium, (2) the ability to control the discretization of the eigendeformation modes at multiple scales to maintain desired accuracy, and (3) hierarchical solution strategy that requires sequential solution of single‐scale problems. A two‐scale formulation is verified against an one‐dimensional model problem for which an analytical solution can be obtained and a three‐scale formulation is validated against tube crash experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

An effective numerical method for solving viscous-inviscid interaction problems

This paper presents a new numerical method to solve the equations of the asymptotic theory of separated flows. A number of measures was taken to ensure fast convergence of the iteration procedure, which is employed to treat the nonlinear terms in the governing equations. Firstly, we selected carefully the set of variables for which the nonlinear finite difference equations were formulated. Secondly, a Newton-Raphson strategy was applied to these equations. Thirdly, the calculations were facilitated by utilizing linear approximation of the boundary-layer equations when calculating the corresponding Jacobi matrix.The performance of the method is illustrated, using as an example, the problem of laminar two-dimensional boundary-layer separation in the flow of an incompressible fluid near a corner point of a rigid body contour. The solution of this problem is non-unique in a certain parameter range where two solution branches are possible.     

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

The optimal incremental function approximation method is implemented for the adaptive and meshless solution of differential equations. The basis functions and associated coefficients of a series expansion representing the solution are selected optimally at each step of the algorithm according to appropriate error minimization criteria. Thus, the solution is built incrementally. In this manner, the computational technique is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a posteriori error estimates. Since the basis functions are associated with the nodes only, the method can be viewed as a meshless method. Variational principles are utilized for the definition of the objective function to be extremized in the associated optimization problems. Complicated data structures, expensive remeshing algorithms, and systems solvers are avoided. Computational efficiency is increased by using low-order local basis functions and the parallel direct search (PDS) optimization algorithm. Numerical results are reported for both a linear and a nonlinear problem associated with fluid dynamics. Challenges and opportunities regarding the use of this method are discussed.     

Error bounds on block Gauss–Seidel solutions of coupled multiphysics problems

Mathematical models in many fields often consist of coupled sub‐models, each of which describes a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub‐models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non‐linear coupled fluid‐temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration. Copyright © 2011 John Wiley & Sons, Ltd.     

The indirect design of fluid channels using a global variational method with trial functions not satisfying the prescribed boundary conditions

The indirect design problem for compressible fluid flow in a channel determines the physical shape of the channel from the prescribed flow on the channel boundary. In this paper, a global variational method for the solution of this problem is described. The non-linear equations are solved by iteration; at each step of this iteration the solution to the equations (which are now linear) is found by a global variational method. The variational method differs from conventional methods in that the trial function does not satisfy the prescribed boundary conditions, but the method reproduces the conditions on the boundary. Results are presented for various physical quantities associated with the channel. Some comments are made on the extensions of these methods to supersonic flow and to the realistic cascade design problem.