Werner C. Rheinboldt

• PhD
• Professor Emeritus at University of Pittsburgh

An overview of the basic local convergence theory of iterative processes for solving non-linear systems of equations is given with applications to some of the principal computational methods, including Newton's method and various of its related forms, and inexact New-ton methods. Then parametrized systems of equations and computational methods for...

In this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development, notably homotopy and degree theory, simplicial complexes and mappings, submanifolds defined by submersions, and singularity and foldpoint theory.

Over the past two decades, both the theory and the numerical analysis of differential algebraic equations (DAEs) have received considerable attention. The interest in such systems has been sustained by the increasing awareness of their pervading relevance in science and engineering. In fact, it is now widely recognized that many physical problems,...

Multibody systems are considered which involve combinations of rigid and elastic bodies. Discretizations of the PDEs, describing the elastic members, lead to a semidiscrete system of ODEs or DAEs. Asymptotic methods are introduced which provide a theoretical basis of various known engineering results for the ODE case. These results are then extende...

Recently, the author introduced a package of algorithms, called MANPAK, for effective computations on implicitly defined submanifolds of Rn. Here algebraically explicit differential algebraic equations (DAEs) are considered; that is, DAEs in which either the algebraic equations and/or the algebraic variables are explicitly specified. Existence proo...

Mathematical models often involve differentiable manifolds that are implicitly defined as the solution sets of systems of nonlinear equations. The resulting computational tasks differ considerably from those arising for manifolds defined in parametric form. Here a collection of algorithms is presented for performing a range of essential tasks on ge...

: Part 1 of this paper presented a theory of distribution solutions of semilinear differential-algebraic equations (DAE's). In particular, it was shown that uniqueness of solutions of initial value problems breaks down completely in the class of discontinuous solutions. Here a mathematical procedure is introduced for selecting physically acceptable...

Existence and uniqueness results are proved for initial-value problems associated with linear, time-varying, differential-algebraic equations. The right-hand sides are chosen in a space of distributions allowing for solutions exhibiting discontinuities as well as impulses. This approach also provides a satisfactory answer to the problem of inconsis...

A coordinate-free reduction procedure is developed for linear time-dependent differential-algebraic equations that transforms their solutions into solutions of smaller systems of ordinary differential equations. The procedure applies to classical as well as distribution solutions. In the case of analytic coefficients the hypotheses required for the...

For minimization problems with nonlinear equality constraints, various numerical tools are shown to become available when the constraint set has a manifold structure. In appropriate local coordinate systems these tools permit the computation, e.g., of the gradient and Hessian of the transformed (unconstrained) objective function. This opens up a ne...

A posteriori error estimators for finite element solutions of multi-parameter nonlinear partial differential equations are based on an element-by-element solution of local linearizations of the nonlinear equation. In general, the associated bilinear form of the linearized problems satisfies a Garding-type inequality. Under appropriate assumptions i...

The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order...

Three recently developed methods for solving the Euler-Lagrange equations are considered, and for their implementations in the codes ELXR5, MDOP5, and MEXX a performance analysis is presented using two realistic simulation problems as test cases.

A method is presented for the computation of a simplicial approximation covering a specified subset Mo of a two-dimensional manifold M in R(n) defined implicitly as the solution set of a nonlinear system F(x) = 0 of n - 2 equations in n unknowns. The given subset M(0) subset of M is the intersection of M with some polyhedral domain in R(n) and is a...

Part 1 of this paper presented a theory of distribution solutions of semilinear differential-algebraic equations (DAE's). In particular, it was shown that uniqueness of solutions of initial value problems breaks down completely in the class of discontinuous solutions. Here a mathematical procedure is introduced for selecting physically acceptable s...

There is strong physical evidence that a full treatment of differential-algebraic equations should be incorporate solutions with jump discontinuities. It is shown here that for semilinear problems the setting of distributions allows for the development of a theory where indeed such discontinuities may occur. This approach also settles the problem o...

A differential-geometric approach for proving the existence and uniqueness of implicit differential-algebraic equations is presented. It provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems. The differential-algebraic equation is transform...

Recently the authors developed a global reduction procedure for linear, time-dependent DAE that transforms their solutions of smaller systems of ODE's. Here it is shown that this reduction allows for the construction of simple, convergent finite difference schemes for such equations.

We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the van...

This paper presents a mathematical characterization of the impasse points of quasilinear DAE's A(x) = G(x), where then A(x) is a nxn matrix having constant but not full r<n in the domain of interest. We show that a reduction procedure permits to reduce the DAE to a quasilinear ODE Al(xi)xi'=G1(xi) Rr and that impasse points correspond to special si...

In an earlier paper (ZAMM 63, 1983, 21), J.P.Fink and W.C.Rheinboldt developed a priori local error estimates for the scalar-parameter case of the reduced basis method by considering the method in a differential-geometric setting. Here it is shown that an analogous setting can be used for the analysis of the method applied to problems with a multid...

We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the van...

The differential equation on a manifold (DEM) method is applied to the equilibrium and rate sensitive constitutive equations governing axisymmetric punch stretching. This method is a variant of the mixed finite element method, and finite elements are used to approximate the displacements, pressure, plastic strains and the effective strain. The resu...

A new approach to the construction of a posteriori error estimates for finite element solutions of multiparameter nonlinear problems presented. The estimates are derived from local, element-by-element solutions of linearizations of the problems; they turn out to be very effective, computationally rather inexpensive, and insensitive to the choice of...

An algorithm is presented for the computation of the second fundamental tensorV of a Riemannian submanifoldM ofR n . FromV the riemann curvature tensor ofM is easily obtained. Moreover,V has a close relation to the second derivative of certain functionals onM which, in turn, provides a powerful new tool for the computational determination of multi...

This paper presents a general existence and uniqueness theory for differential-alegebraic equations extending the well known ODE theory. Both local and global aspects are considered, and the definition of the index for nonlinear problems is elucidated. For the case of linear problems with constant coefficients the results are shown to provide an al...

A differential-geometric approach for the numerical solution of mixed differential-algebraic systems of equations is presented and a general local parametrization for such systems is constructed. Multistep ODE solvers are then applied for obtaining locally a numerical approximation to the solution of the differential-algebraic system. The class of...

A theorem of S.Smale [4] on the convergence of Newton's method for analytic functions holds only for mappings which are analytic on the entire space. The result is generalized to functions which are analytic on an open subset and, for the original case, when this subsetis the entire space the controlling constant given by Smale is improved.

A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR n toR m ,p=n–m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of th...

The goal of mathematical modelling of sheet metal forming processes is to provide predictive tools for use in the design of stamping processes and the selection of sheet materials. Most current approaches to finite element modelling of large deformation, elastic–plastic sheet metal forming problems use a rate form of the virtual work (equilibrium)...

While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter-dependent equation F(z, lambda ) equals 0. The prese...

This paper presents a framework for studying singular points on the solution manifold of a parameter-dependent nonlinear equation $F(z,\lambda ) = 0$. The approach is based on a systematic combination of general constrained mappings with the tangent map of differential geometry. This framework is then used to develop a geometrically instructive and...

Many current approaches to finite element modelling of large deformation elastic—plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three-field formulation in which displacements, stresses an...

New sufficient conditions for the monotone convergence of Newton's method for solving nonlinear systems of equations are given. These conditions are affine-invariant and less restrictive than the hypothesis of Baluev's theorem.Neue hinreichende Bedingungen fr die monotone Konvergenz der Newton-Methode fr die Lsung nichtlinearer Gleichung werden geg...

An abstract is not available.

The paper describes an experimental software system for the adaptive solution of a class of non-linear boundary value problems. The system is a further development of an earlier adaptive solver of linear elliptic problems. It retains the functionality of the earlier program but incorporates new a posteriori error estimates and a continuation proced...

The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y0 involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds...

: Many applications lead to nonlinear, parameter dependent equations H(y,t) = y sub o, where H: Y x T yields Y, y sub o epsilon rge H, and the state space Y is infinite-dimensional while the parameter space T has finite dimension. The case dim T = 1 is of special interest in connection with continuation methods. For this case, a general theory is d...

The design of a package of continuation procedures called PITCON to handle the following tasks is described: (1) follow numerically any a priori specified curve on an equilibrium manifold; (2) on such a curve determine the exact location of target points where a given variable has a specified value; and (3) on such a curve identify and compute exac...

For a general assessment of the structure, qualitative results about the displacements, stress-distributions, etc. are needed with an engineering accuracy of, say 10-20%. On the other hand, the indicated decisions are based usually on relatively few data items with a higher accuracy, say, in the range of 2-5%, such as, the displacements or stresses...

In the study of many equilibrium problems it is important to determine the location of turning points which may signify a loss of stability. A number of algorithms is known for the computation of such points. In this paper experimental comparisons are presented of a total of sixteen variations of ten such methods. For this a set of seven test probl...

In earlier papers the authors introduced and analyzed the calculation of reliable, a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them. While most of this work concerned linear problems, this paper is intended to show that the same approaches remain also highly ef...

The study of various equilibrium phenomena leads to nonlinear equations (1) $F(y,u) = 0$, where $y \in R^n $ is a vector of behavior or state variables, $u \in R^p $ a vector of $p \geqq 2$ parameters or controls and $F:D \subset R^n \times R^p \to R^n $ a sufficiently differentiable map. The solution set of (1) in $R^n \times R^p $ is often called...

Continuation methods are considered here in a broad sense as the collection of methods needed for the computational analysis of specified parts of the solution field of “under-determined” equations Fx = c where F: Rm → Rn, m >; n. is given and any suitable m−n of the variables x, are designated as parameters. Such equations arise frequently in stru...

The paper develops a theory of a posteriors error estimates under the $L_p $-energy norm for $2 \leqq p \leqq \infty $. The theory is based on a general concept of error indicators and error estimators. Several specific examples of these quantities are introduced and analyzed in detail. The results provide a variety of easily computable error estim...

The nature of adaptive processes is reviewed using as an example a specific finite element problem. Several possible formulations for the objective of mesh refinement processes are given. For the most natural of these objectives the A*-algorithms of artificial intelligence turn out to provide a solution process which is known to be optimal in a cer...

A general labeled tree structure is introduced for a class of nonuniform two-dimensmnal fimte element meshes After defimtmn of the bamc structure, the fundamental access algomthms on the tree are presented m detail. Then the principal algorithms needed for the fimte element computatmns are discussed, including the refinement of the mesh, the comput...

The design for a novel prototype finite-element system IS presented which meets the followmg four goals (a) the system constitutes an application-independent finite-element solver for a certam class of linear elhptic problems based on a weak mathematical formulation, (b) it incorporates extensive adaptive approaches to mimmize the critical decision...

The principal aspects are outlined of a prototype, adaptive finite element system for the solution of a class of linear, elliptic problems defined by a weak mathematical formulation. The system uses results about computable a-posteriori estimates and asymptotically optimal meshes developed earlier by the authors to control the adaptive mesh refinem...

In continuation of earlier work on the graph algorithmic language GRAAL, a new type of graph representation is introduced involving solely the arcs and their incidence relations. In line with the set theoretical foundation of GRAAL, the are graph structure is defined in terms of four Boolean mappings over the power set of the ares. A simple data st...

In this note two new proofs are given of the following characterization theorem of M. Fiedler: Let Cn, n⩾2, be the class of all symmetric, real matrices A of order n with the property that rank (A + D) ⩾ n - 1 for any diagonal real matrix D. Then for any A ε Cn there exists a permutation matrix P such that PAPT is tridiagonal and irreducible.

The growing influence of modern electronic computing in many fields of knowledge has contributed to a dramatic increase and diversification in the application of mathematics to other disciplines. No longer are the uses of mathematics confined exclusively to the physical sciences and engineering; they are found with increasing frequency in the socia...

An algorithmic language, GRAAL, is defined, as an extension of ALGOL 60 (Revised), for describing and implementing graph algorithms of the type arising in applications. It is based on a set algebraic model of graph theory which defines the graph structure in terms of user specified morphisms between certain set algebraic structures over the node an...

In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton’s method results in the same linear algebraic system of equations as application of Newton's method followed by discretization. This leads to the general problem of determining when the two frequently used operations of discretiza...