automatic energy conserving space time refinement for linear dynamic structural problems in this paper a local space time automatic refinement method star method is developed to efficiently solve time dependent problems using fem techniques the automatic process is driven by an energy or a displacement error indicator which controls the precision of the result the star method solves the numerical problem on grids with different mesh size for the newmark schemes a general demonstration using the energy method gives the interface conditions between two successive grids which is compatible with the stability of the scheme finally using a linear one dimensional example the convergence of the method and the precision of the results are discussed copyright 2005 john wiley sons ltd

an exact algorithm for higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds and other heuristics we develop a two level tree search algorithm for solving higher dimensional packing problems to optimality computational results are reported including optimal solutions for all two dimensional test problems from recent literature this is the third in a series of articles describing new approaches to higher dimensional packing

binary spatial partitioning of the central difference time integration scheme for explicit fast transient dynamics this paper proposes a generalization of the explicit central difference time integration scheme using a time step variable not only in time but also in space the solution at each element node is advanced in time following local rather than global stability limitations this allows substantial saving of computer time in realistic applications with non uniform meshes especially in multi field problems like fluid structure interactions a binary scheme in space is used time steps are not completely arbitrary but stay in a constant ratio of two when passing from one partition level to the next one this choice greatly facilitates implementation via an integer based logic ensures inherent synchronization and avoids any interpolations necessary in other partitioning schemes in the literature but which may reduce numerical stability the mesh partition is automatically built up and continuously updated by simple spatial adjacency considerations the resulting algorithm deals automatically with large variations in time of stability limits the paper introduces the core spatial partitioning technique in the lagrangian formulation some academic numerical examples allow a detailed comparison with the standard spatially uniform algorithm a final more realistic example shows the application of partitioning in simulations with arbitrary lagrangian eulerian formulation and fully coupled boundary conditions fluid structure interaction copyright 2008 john wiley sons ltd

an explicit dynamics extended finite element method part 2 element by element stable explicit explicit dynamic scheme abstract this paper presents a general explicit time integration scheme for dynamics simulations using the extended finite element method with standard critical time step we use the generalized mass lumping technique proposed in part i of this paper this technique allows us to consider arbitrary enrichment functions in the x fem for explicit dynamics simulations in this second part the proposed approach allows the use of standard finite elements critical time step estimates for that purpose we develop a classical element by element strategy that couples the standard central difference scheme with the unconditionally stable explicit scheme proposed by chang s y chang an explicit method with improved stability property int j numer methods engrg 77 8 2008 1100 1120 this scheme coupled with x fem allows us to recover fe critical time steps independently of the enrichment functions considered furthermore a study of the stability property of this new explicit scheme is proposed in the x fem framework both with fixed enrichments and evolving enrichments with time some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given

stability of asynchronous variational integrators the adoption of multiple time step integrators can provide substantial computational savings for mechanical systems with multiple time scales however the scope of these savings may be limited by the range of allowable time step choices in this paper we analyze the linear stability of the fully asynchronous methods termed avi for asynchronous variational integrators we perform a detailed analysis for the case of a one dimensional particle moving under the action of a soft and a stiff quadratic potential integrated with two time steps in rational ratios in this case we provide sufficient conditions for the stability of the method these generalize to the fully asynchronous avi case the results obtained for synchronous multiple time stepping schemes such as r respa which show resonances when the larger time step is a multiple of the effective half period of the stiff potential additionally we numerically investigate the appearance of instabilities based on the experimental observations we conjecture the existence of a dense set of unstable time steps when arbitrary rational ratios of time steps are considered in this way unstable schemes for arbitrarily small time steps can be obtained however the vast majority of these instabilities are extremely weak and do not present an obstacle to the use of these integrators we then applied these results to analyze the stability of multiple time step integrators in the more complex mechanical systems arising in molecular dynamics and solid dynamics we explained why strong resonances are ubiquitously found in the former while rarely encountered in the latter finally in this paper we introduce a formulation of avi that highlights the symplectic nature of the algorithm complementing those introduced earlier by other authors

application of triangular space time finite elements to problems of wave propagation abstract the displacement hp space time formulation derived from hamilton s weak principle hwp is studied for simultaneous space time solutions without conventional semi discretizations a special type of discretization of space time is presented in which both space and time dependency are removed the theoretical basis for a displacement hp version space time finite element is developed and a bilinear formulation of space time dynamics is presented in which numerical studies can be performed for h or p version elements of any shape the commitment to a weak formulation of force and momentum allows appropriate solution discontinuities to develop naturally at element boundaries to predict the steep gradients of stress in wave like motions triangular elements are used to cut across space and time such that discontinuities in force or momentum can be located at the element boundary a unified theory of shape functions for space time finite elements is developed and the shape functions for triangular and parallelogram elements are derived in terms of a set of orthogonal functions the assembly procedure for hp version space time finite elements is also discussed to speed the rate of convergence an extraction technique is developed to obtain forces and momenta from integrals of generalized displacements not from their derivatives

lp bounds in an interval graph algorithm for orthogonal packing feasibility we consider the feasibility problem opp orthogonal packing problem in higher dimensional orthogonal packing given a set of d dimensional d 2 rectangular items decide whether all of them can be orthogonally packed in the given rectangular container without rotation the one dimensional 1d bar lp relaxation of opp reduces the latter to a 1d cutting stock problem where the packing of each stock bar represents a possible 1d stitch through an opp layout the dual multipliers of the lp provide us with another kind of powerful bounding information conservative scales we investigate how the set of possible 1d packings can be tightened using the overlapping information of item projections on the axes with the goal to tighten the relaxation we integrate the bar relaxation into an interval graph algorithm for opp which operates on such overlapping relations numerical results on 2d and 3d instances demonstrate the efficiency of tightening leading to a speedup and stabilization of the algorithm

an exact algorithm for the two dimensional strip packing problem this paper considers the two dimensional strip packing problem 2sp in which a set of rectangular items have to be orthogonally packed without overlapping into a strip of a given width and infinite height by minimizing the overall height of the packing the 2sp is np hard in the strong sense and finds many practical applications we propose reduction procedures lower and upper bounds and an exact algorithm for the 2sp the new lower bounds are both combinatorial bounds and bounds derived from different relaxations of mathematical formulations of the 2sp the new upper bounds are obtained by constructive heuristics based on different strategies to place the items into the strip the new exact method is based on a branch and bound approach computational results on different sets of test problems derived from the literature show the effectiveness of the new lower and upper bounds and of the new exact algorithm

minimal graphic functions on manifolds of nonnegative ricci curvature we study minimal graphic functions on complete riemannian manifolds with nonnegative ricci curvature euclidean volume growth and quadratic curvature decay we derive global bounds for the gradients for minimal graphic functions of linear growth only on one side then we can obtain a liouville type theorem with such growth via splitting for tangent cones of at infinity when in contrast we do not impose any growth restrictions for minimal graphic functions we also obtain a liouville type theorem under a certain nonradial ricci curvature decay condition on in particular the borderline for the ricci curvature decay is sharp by our example in the last section 2014 wiley periodicals inc

stability of explicit subcycling time integration with linear interpolation for first order finite element semidiscretizations an explicit subcycling time integration algorithm based on an element partition is presented for first order finite element systems this method uses linear interpolation to compute intermediate values at nodes in the subdomains integrated with the larger time steps stability of the algorithm for the two partition cases is analyzed by showing that the eigenvalues of the amplification matrix for the total integration cycle are not greater than one in absolute value using this stability criterion critical time steps for the element subdomains are derived in terms of element eigenvalues several numerical examples are considered to compare the accuracy of the method with previously developed subcycling algorithms to numerically verify the stability analysis

easily computable facets of the knapsack polytope it is known that facets and valid inequalities for the knapsack polytope p can be obtained by lifting a simple inequality derived from each minimal cover we study the computational complexity of such lifting in particular we show that the task of computing a lifted facet can be accomplished in o ns where s n is the cardinality of the minimal cover also for a lifted inequality with integer coefficients we show that the dual tasks of recognizing whether the inequality is valid for p or is a facet of p can be done within the same time bound

analysis and implementation of a new constant acceleration subcycling algorithm an algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps with integer time step ratios the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle the algorithm is shown to have good accuracy and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms the implementation of a generalised form of the algorithm with non integer time step ratios is presented c 1997 by john wiley sons ltd

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

multi time step and two scale domain decomposition method for non linear structural dynamics in this paper we propose a method to improve the means of taking into account the specific time scale and space scale characteristics in time dependent non linear problems this method enables the use of arbitrary time steps in each subdomain these can be coupled by prescribing continuous velocities at the interfaces which are modelled using a dual schur formulation for certain subdomains in space we adopt a two scale resolution technique inspired by the multigrid methods in order to obtain the part of the solution related to small variation lengths on a refined scale and the part corresponding to large variation lengths on a coarse scale for non linear problems we propose an algorithm with a single iteration level to deal with both the non linear equilibrium and the two space scales thanks to a two grid method in which the relaxation steps are performed using a non linear preconditioned conjugate gradient algorithm finally we present an example which demonstrates the feasibility of the method copyright 2003 john wiley sons ltd

a multi timestep robin robin domain decomposition method for time dependent advection diffusion problems abstract many physical processes present separated evolution time scales in different regions such as windblown sand in the desert the evolution of bed surface locally affects the sandy wind flow which erodes the sand in turns while few meters higher from the ground the airflow is almost stationary this and other applications suggest to improve the overall computational efficiency by locally adapting the time advancing step to the evolution in each region we propose a multi timestep robin robin domain decomposition method which uses mutually proportional time steps based on a predictor corrector strategy coupled with linear interpolation in time the resulting algorithm is completely segregated several numerical tests obtained by the finite volume code openfoam are reported to show its stability and convergence properties

enabling fast stable and accurate peridynamic computations using multi time step integration abstract peridynamics is a nonlocal extension of classical continuum mechanics that is well suited for solving problems with discontinuities such as cracks this paper extends the peridynamic formulation to decompose a problem domain into a number of smaller overlapping subdomains and to enable the use of different time steps in different subdomains this approach allows regions of interest to be isolated and solved at a small time step for increased accuracy while the rest of the problem domain can be solved at a larger time step for greater computational efficiency performance of the proposed method in terms of stability accuracy and computational cost is examined and several numerical examples are presented to corroborate the findings

an exact algorithm for the two dimensional orthogonal packing problem with unloading constraints this paper describes an exact algorithm for solving a two dimensional orthogonal packing problem with unloading constraints which occurs as a subproblem of mixed vehicle routing and loading problems the packing considered in this work is basically a feasibility problem involving a single bin the problem is addressed through a decomposition approach wherein a branch and cut algorithm is designed for solving a one dimensional relaxation of the original problem when an integer solution is found in the branching tree a subsidiary problem is solved to identify a two dimensional packing that does not lead to any overlap and satisfies the unloading constraints cuts are added when the subsidiary problem proves to be infeasible several preprocessing techniques aimed at reducing the size of the solution space and uncovering infeasibility are also described a numerical comparison with the best known exact method is reported at the end based on benchmark instances

a monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics a new hybrid multi time method for multi time scales structural dynamics simulations is described a monolithic method in a schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain in the proposed method zero numerical dissipation is ensured at the interface this implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface for that purpose velocity continuity is ensured in a weak sense at the interfaces and time integrators newmark hht simo krenk verlet are introduced in a unified framework incremental velocity formulation furthermore dynamics governing equations are introduced from a weak formulation in time in other words equilibrium equation is no more ensured in a strong sense at a given time step but rather on average on a time interval some numerical examples illustrate the efficiency and the robustness of the proposed method for ratio of time scales close to 1000 without any numerical dissipation at the interfaces

asynchronous multi domain variational integrators for non linear problems we develop an asynchronous time integration and coupling method with domain decomposition for linear and non linear problems in mechanics to ensure stability in the time integration and in coupling between domains we use variational integrators with local lagrange multipliers to enforce continuity at the domain interfaces the asynchronous integrator lets each domain step with its own time step using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed we show that in practice the time step is limited by accuracy requirements rather than by stability requirements copyright 2008 john wiley sons ltd

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

exact algorithms for the two dimensional strip packing problem with and without rotations we propose exact algorithms for the two dimensional strip packing problem 2sp with and without 90 rotations we first focus on the perfect packing problem pp which is a special case of 2sp wherein all given rectangles are required to be packed without wasted space and design branch and bound algorithms introducing several branching rules and bounding operations a combination of these rules yields an algorithm that is especially efficient for feasible instances of pp we then propose several methods of applying the pp algorithms to 2sp our algorithms succeed in efficiently solving benchmark instances of pp with up to 500 rectangles and those of 2sp with up to 200 rectangles they are often faster than existing exact algorithms specially tailored for problems without rotations

asymptotic dirichlet problems in warped products we study the asymptotic dirichlet problem for killing graphs with prescribed mean curvature h in warped product manifolds m times varrho mathbb r in the first part of the paper we prove the existence of killing graphs with prescribed boundary on geodesic balls under suitable assumptions on h and the mean curvature of the killing cylinders over geodesic spheres in the process we obtain a uniform interior gradient estimate improving previous results by dajczer and de lira in the second part we solve the asymptotic dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to infty provided that h satisfies certain bounds with respect to the sectional curvatures of m and the norm of the killing vector field finally we obtain non existence results if the prescribed mean curvature function h grows too fast

a combinatorial characterization of higher dimensional orthogonal packing higher dimensional orthogonal packing problems have a wide range of practical applications including packing cutting and scheduling previous efforts for exact algorithms have been unable to avoid structural problems that appear for instance in two or higher dimensional space we present a new approach for modeling packings using a graph theoretical characterization of feasible packings our characterization allows it to deal with classes of packings that share a certain combinatorial structure instead of having to consider one packing at a time in addition we can make use of elegant algorithmic properties of certain classes of graphs this allows our characterization to be the basis for a successful branch and bound framework this is the first in a series of papers describing new approaches to higher dimensional packing

multi time step integration using nodal partitioning an algorithm is presented which integrates different groups of nodes of a finite element mesh with different time steps and different integrators since the nodal groups are updated independently no unsymmetric systems need be solved stability is demonstrated by showing that an energy norm of the solution decreases after every update if the time step is less than a given critical value the element eigenvalue inequality theorem is used to give the critical time step in terms of element eigenvalues

existence and non existence of minimal graphic and p harmonic functions we prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete riemannian manifold m with only one end if m has asymptotically non negative sectional curvature on the other hand we prove the existence of bounded non constant minimal graphic and p harmonic functions on rotationally symmetric cartan hadamard manifolds under optimal assumptions on the sectional curvatures

a study of the stability of subcycling algorithms in structural dynamics algorithms for explicit integration of structural dynamics problems with multiple time steps subcycling are investigated only one such algorithm due to smolinski and sleith has proved to be stable in a classical sense a simplified version of this algorithm that retains its stability is presented however as with the original version it can be shown to sacrifice accuracy to achieve stability another algorithm in use is shown to be only statistically stable in that a probability of stability can be assigned if appropriate time step limits are observed this probability improves rapidly with the number of degrees of freedom in a finite element model the stability problems are shown to be a property of the central difference method itself which is modified to give the subcycling algorithm a related problem is shown to arise when a constraint equation in time is introduced into a time continuous space time finite element model

two feti based heterogeneous time step coupling methods for newmark and α schemes derived from the energy method abstract the purpose of this paper is to describe a general methodology to obtain multi time step coupling methods for different time integration schemes newmark hht wbz ch on the basis of energetic considerations the general framework of the feti method is considered for building the heterogeneous time step coupling methods by ensuring the gluing of subdomains with lagrange multipliers the proposed methods for coupling subdomains with different time integration schemes and time steps fine and large time steps are based on the pseudo energy measure as introduced by hughes in the so called energy method employed for proving the stability of implicit explicit algorithms assuming a given variation of the lagrange multipliers over the large time step kinematic conditions are derived by ensuring the zero pseudo energy at the interface two methods will be derived in this paper the first new method for which the interface problem involving the lagrange multipliers is solved at the large time scale can handle the popular dissipative schemes hht wbz ch in the particular case of the newmark schemes the new method matches with the ph method proposed by prakash and hjelmstad the second new method is based at the micro time scale it can be viewed as an extension of the method proposed by gravouil and combescure gc method for the coupling of the schemes nonetheless like the gc method the new micro time based method turns out to be dissipative when non dissipative second order accurate schemes are involved and looses one order of accuracy

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

a multi time step integration algorithm for structural dynamics based on the modified trapezoidal rule an explicit multi time step subcycling integration method is proposed for solving semi discretized structural dynamics problems based on nodal partition theory this method is derived from the modified trapezoidal rule method mtm to achieve numerical stability virtually fixed physical quantities displacement and velocity are assumed for the non updated nodal groups the energy method is used to analyze the stability of the new algorithm and the critical time step for each nodal group is determined in terms of the maximum frequency of the elements in the associated subdomain some example problems are evaluated to numerically examine the accuracy and stability of the algorithm

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

notes on the stability of non rectangular space time finite elements the method of space time finite elements enables the simple solution of quite new problems it is possible to assume the arbitrary partition of the structure area in each moment of integration of the motion equation instability is caused by a too large time step and too great changes of joint locations in successive time steps a changeable spatial partition is useful in contact dynamic problems in the case of a travelling support generally in problems with movable edges in this paper a stability problem is described and some investigations for chosen types of non rectangular space time finite elements are carried out linear and surface elements which in time space gain an additional time dimension are presented some numerical examples prove the efficiency of the described method under determined limitations

a partial velocity approach to subcycling structural dynamics subcycling or the use of different timesteps at different nodes can be an effective way of improving the computational efficiency of explicit transient dynamic structural solutions the method that has been most widely adopted uses a nodal partition extending the central difference method in which small timestep updates are performed interpolating on the displacement at neighbouring large timestep nodes this approach leads to narrow bands of unstable timesteps or statistical stability it also can be in error due to lack of momentum conservation on the timestep interface the author has previously proposed energy conserving algorithms that avoid the first problem of statistical stability however these sacrifice accuracy to achieve stability an approach to conserve momentum on an element interface by adding partial velocities is considered here applied to extend the central difference method this approach is simple and has accuracy advantages the method can be programmed by summing impulses of internal forces evaluated using local element timesteps in order to predict a velocity change at a node however it is still only statistically stable so an adaptive timestep size is needed to monitor accuracy and to be adjusted if necessary by replacing the central difference method with the explicit generalized alpha method it is possible to gain stability by dissipating the high frequency response that leads to stability problems however coding the algorithm is less elegant as the response depends on previous partial accelerations extension to implicit integration is shown to be impractical due to the neglect of remote effects of internal forces acting across a timestep interface c 2002 elsevier science b v all rights reserved

a new exact method for the two dimensional orthogonal packing problem abstract the two dimensional orthogonal packing problem 2 opp consists in determining if a set of rectangles items can be packed into one rectangle of fixed size bin in this paper we propose two exact algorithms for solving this problem the first algorithm is an improvement on a classical branch bound method whereas the second algorithm is based on a new relaxation of the problem we also describe reduction procedures and lower bounds which can be used within enumerative methods we report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods the second method is competitive compared to the best previous methods it can be seen that our new relaxation allows an efficient detection of non feasible instances

potential theory for manifolds with boundary and applications to controlled mean curvature graphs in this paper we characterize the neumann parabolicity of manifolds with boundary in terms of a new form of the classical ahlfors maximum principle and of a version of the so called kelvin nevanlinnaroyden criterion the motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a riemannian product of the type n r in this direction two kind of results will be presented height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold which extend results by a ros and h rosenberg valid for domains of r and slice type results for graphs whose superlevel sets have finite volume finally the use of the ahlfors maximum principle allows us to establish a connection between the neumann parabolicity and the dirichlet parabolicity commonly used in minimal surface theory in particular we will be able to give a deterministic proof of special cases of a result by r neel

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

half space theorem embedded minimal annuli and minimal graphs in the heisenberg group we construct a one parameter family of properly embedded minimal annuli in the heisenberg group nil3 endowed with a left invariant riemannian metric these annuli are not rotationally invariant this family of annuli is used to prove a vertical half space theorem which is then applied to prove that each complete minimal graph in nil3 is entire also it is shown that the sister surface of an entire minimal graph in nil3 is an entire constant mean curvature cmc 1 graph in h 2 r and vice versa this gives a classification of all entire cmc 1 graphs in h 2 r finally we construct properly embedded cmc 1 annuli in h 2 r