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  • 251. El-Gabry, Lamyaa
    et al.
    Saha, Ranjan
    KTH, School of Industrial Engineering and Management (ITM), Energy Technology, Heat and Power Technology.
    Fridh, Jens
    KTH, School of Industrial Engineering and Management (ITM), Energy Technology, Heat and Power Technology.
    Fransson, Torsten
    KTH, School of Industrial Engineering and Management (ITM), Energy Technology, Heat and Power Technology.
    Measurements of Hub Flow Interaction on Film Cooled Nozzle Guide Vane in Transonic Annular Cascade2015In: Journal of turbomachinery, ISSN 0889-504X, E-ISSN 1528-8900, Vol. 137, no 8, article id 081004Article in journal (Refereed)
    Abstract [en]

    An experimental study has been performed in a transonic annular sector cascade of nozzle guide vanes (NGVs) to investigate the aerodynamic performance and the interaction between hub film cooling and mainstream flow. The focus of the study is on the endwalls, specifically the interaction between the hub film cooling and the mainstream. Carbon dioxide (CO2) has been supplied to the coolant holes to serve as tracer gas. Measurements of CO2 concentration downstream of the vane trailing edge (TE) can be used to visualize the mixing of the coolant flow with the mainstream. Flow field measurements are performed in the downstream plane with a five-hole probe to characterize the aerodynamics in the vane. Results are presented for the fully cooled and partially cooled vane (only hub cooling) configurations. Data presented at the downstream plane include concentration contour, axial vorticity, velocity vectors, and yaw and pitch angles. From these investigations, secondary flow structures such as the horseshoe vortex, passage vortex, can be identified and show the cooling flow significantly impacts the secondary flow and downstream flow field. The results suggest that there is a region on the pressure side (PS) of the vane TE where the coolant concentrations are very low suggesting that the cooling air introduced at the platform upstream of the leading edge (LE) does not reach the PS endwall, potentially creating a local hotspot.

  • 252.
    Elmukashfi, Elsiddig
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Experimental investigations of crack propagation in rubber under dif-ferent loading rates, temperatures and fracture modes2015Report (Other academic)
    Abstract [en]

    In the present paper, the fracture behavior of carbon-black natural rubber material is experimentally studied. The cracked pure shear and the single edge notch specimens were used for investigating both pure mode I and mixed mode I and II fracture behavior, respectively. Further, different testing conditions were employed in the case of the cracked pure shear specimens. The specimens were subjected to three different loading rates and they were tested in two different temperatures. For studying the crack growth, a high speed camera at up to 7000 frames/s was used to follow the progress of the crack and later a post-processor was used to obtain the crack trajectory and velocity at different stages. The method introduced previously by the present author (Elmukashfi in report 580, Department of Solid Mechanics, Royal Institute of Technology (KTH), 2015) was used to obtain the critical tearing energy using the cracked pure shear specimens. Hence, the uncracked pure shear specimens were subjected to cyclic loading history in order to obtain the true elastic energy in pure shear. The single edge notch specimens were tested in room temperature under quasi-static loading. The pure mode I results suggest that the critical tearing depends strongly on the loading rate as well as the temperature. The tearing behavior shows stick-slip pattern at low tearing rates and smooth propagation at high velocities. The size of the stick-slip region is reduced significantly by increasing the loading rate as well as the temperature. In the mixed mode I and II, the transition from the stick-slip to smooth propagation and the transition from mixed mode I and II take place approximately simultaneously.

  • 253.
    Elmukashfi, Elsiddig
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Modeling of fracture and damage in rubber under dynamic and quasi-static conditions2015Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Elastomers are important engineering materials that have contributed to the different technical developments and applications since the 19th century. The study of crack growth mechanics for elastomers is of great importance to produce reliable products and therefore costly failures can be prevented. On the other hand, it is fundamental in some applications such as adhesion technology, elastomers wear, etc. In this thesis work, crack propagation in rubber under quasi-static and dynamic conditions is investigated.

    In Paper A, theoretical and computational frameworks for dynamic crack propagation in rubber have been developed. The fracture separation process is presumed to be described by a cohesive zone model and the bulk behavior is assumed to be determined by viscoelasticity theory. The numerical model is able to predict the dynamic crack growth. Further, the viscous dissipation in the continuum is found to be negligible and the strength and the surface energy vary with the crack speed. Hence, the viscous contribution in the innermost of the crack tip has been investigated in Paper B. This contribution is incorporated using a rate-dependent cohesive model. The results suggest that the viscosity varies with the crack speed. Moreover, the estimation of the total work of fracture shows that the fracture-related processes contribute to the total work of fracture in a contradictory manner.

    A multiscale continuum model of strain-induced cavitation damage and crystallization in rubber-like materials is proposed in Paper C. The model adopts the network decomposition concept and assumes the interaction between the filler particles and long-chain molecules results in two networks between cross-links and between the filler aggregates. The network between the crosslinks is assumed to be semi-crystalline, and the network between the filler aggregates is assumed to be amorphous with the possibility of debonding. Moreover, the material is assumed to be initially non-cavitated and the cavitation may take place as a result from the debonding process. The cavities are assumed to exhibit growth phase that may lead to complete damage. The comparison with the experimental data from the literature shows that the model is capable to predict accurately the experimental data.

    Papers D and E are dedicated to experimental studies of the crack propagation in rubber. A new method for determining the critical tearing energy in rubber-like materials is proposed in Paper D. The method attempts to provide an accurate prediction of the tearing energy by accounting for the dissipated energy due to different inelastic processes. The experimental results show that classical method overestimates the critical tearing energy by approximately 15%. In Paper E, the fracture behavior of carbon-black natural rubber material is experimentally studied over a range of loading rates varying from quasi-static to dynamic, different temperatures, and fracture modes. The tearing behavior shows a stick-slip pattern in low velocities with a size dependent on the loading rate, temperature and the fracture mode. Smooth propagation results at high velocities. The critical tearing depends strongly on the loading rate as well as the temperature.

  • 254.
    Emil Génetay, Johansen
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Andersson, Aaron
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Analytisk mekanik med datoralgebra2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In this project we have used the theory of analytical mechanics to derive equations of motion for complex mechanical systems in order to study their behaviour over time. Even if the theory provides powerful tools to tackle tough problems which would be very hard to solve by methods of Newtonian mechanics, you are still relatively limited without running any computer simulations. One realises that in a faster rate it will be an incredibly difficult task to find an analytical solution when the degrees of freedom increases. Even if one would succeed to derive the equations, one still has to solve them to get the information of the systems behavior. Of experience one knows that the equations in general are nonlinear and hence must be integrated numericaly. These nonlinearities preposses the sensible dependence on initial conditions which means that small perturbations from equilibrium will lead to divergent outcomes in the continuing evolution of the motion. The motion is then what you call chaotic. Since we have been dealing with nonlinear problems, numerical and symbolical analysis has been performed with the computer aid. Hence, a big part of this project has been to get familiar with the Maple based simulation program ’Sophia’, developed by the department of mechanics at KTH [7]. With this program we could simulate different kinds of pendulum systems and by plotting their solutions, investigate the influence on the overall motion by the parameter values and initial conditions.

  • 255.
    Enflo, Bengt Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Theoretical and Applied Mechanics.
    Hedberg, Claes M.Blekinge Institute of Technology.Kari, LeifKTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, MWL Structural and vibroacoustics.
    Non-Linear Acoustics: Fundamentals and Applications2008Conference proceedings (editor) (Refereed)
  • 256. Ericsson, K. A.
    et al.
    Toll, Staffan
    KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Lightweight Structures.
    Manson, J. -AE.
    The two-way interaction between anisotropic flow and fiber orientation in squeeze flow1997In: Journal of Rheology, ISSN 01486055 (ISSN), Vol. 41, no 3, p. 491-511Article in journal (Refereed)
    Abstract [en]

    The rheology of a discontinuous fiber filled polypropylene in squeeze flow between parallel plates is studied. The material has an initial anisotropic fiber orientation distribution and therefore displays a strongly anisotropic in-plane flow behavior with predominant flow transverse to the axis of principal orientation. The kinematic field is computed using a linear, orthotropic constitutive model, where the fibers are assumed to move affinely with the surrounding fluid. The fiber orientation distribution is updated in each timestep thus coupling orientation and flow. Two different orientation descriptions are evaluated: orientation tensors with closure approximations, and a technique based on direct solution of the orientation of a set of test fibers. The two methods are first compared to exact solutions of the orientation distribution function in simple shear and pure extension; the direct solution is exact within numerical error, wheras the methods based on orientation tensors and quadratic and hybrid closure fail to correctly describe any transient fiber orientation evolution. Finally, the orientation representations are implemented in the kinematic model and compared to the experimental data; the direct solution method is found to give a very accurate prediction of the observed flow kinematics, whereas the other techniques result in substantial errors. © 1997 The Society of Rheology.

  • 257. Ericsson, K. A.
    et al.
    Toll, Staffan
    KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Lightweight Structures.
    Månson, J. -AE.
    Sliding plate rheometry of planar oriented concentrated fiber suspension1997In: Rheologica Acta, ISSN 0035-4511, E-ISSN 1435-1528, Vol. 36, no 4, p. 397-405Article in journal (Refereed)
    Abstract [en]

    The rheology of concentrated planar fiber suspensions is investigated. A new experimental technique for fiber suspensions based on a sliding plate rheometer incorporating a shear stress transducer is developed. It is shown that this instrument works well for the tested material systems. The Theological behavior in steady shear is subsequently investigated. The results can be largely explained by a combination of frictional and hydrodynamic interaction. Despite this evidence of friction no yield stress could be detected for the investigated shear rates. It was also found that the fiber aspect ratio did not influence the steady shear viscosity.

  • 258.
    Ericsson, Max
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Simulating Bird Strike on Aircraft Composite Wing Leading Edge.2012Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this master thesis project the possibility to model the response of a wing when subjected to bird strike using finite elements is analyzed. Since this transient event lasts only a few milliseconds the used solution method is explicit time integration. The wing is manufactured using carbon fiber laminate. Carbon fiber laminates have orthotropic material properties with different stiffness in different directions. Accordingly, there are damage

    mechanisms not considered when using metal that have to be modeled when using composites. One of these damage mechanisms is delamination which occurs when cured layers inside a component become separated. To simulate this phenomenon, multiple layers of shell elements with contact in between are used as a representation of the interface where a component is likely to delaminate.

    By comparing experimental and simulated results the model of delamination is verified and the influence of different parameters on the results is investigated. Furthermore, studies show that modeling delamination layers in each possible layer of a composite stack is not optimal due to the fact that the global stiffness of the laminate is decreased as more layers are modeled. However, multiple layers are needed in order to mitigate the spreading of delamination and obtain realistic delaminated zones.

    As the laminates are comprised of carbon fiber and epoxy sheets it is of importance to include damage mechanisms inside each individual sheet. Accordingly, a composite material model built into the software is used which considers tensile and compressive stress in fiber and epoxy. The strength limits are then set according to experimental test data.

    The bird is modeled using a mesh free technique called Smooth Particle Hydrodynamics using a material model with properties similar to a fluid. The internal pressure of the bird model is linked to the change in volume with an Equation of State. By examining the bird models behavior compared to experimental results it is determined to have a realistic impact on structures.

    A model of the leading edge is then subjected to bird strike according to European standards. The wing skin is penetrated indicating that reinforcements might be needed in order to protect valuable components inside the wing structure such as the fuel tank. However, the results are not completely accurate due to the fact that there is little experimental data

    available regarding soft body penetration of composite laminates. As a consequence, the simulation cannot be confirmed against real experimental results and further investigations are required in order to have confidence in modeling such events. Furthermore, the delamination due to the bird strike essentially spreads across the whole model. Since only one layer of delamination is included the spread is most likely overestimated.

  • 259. Eriksson, A.
    Equilibrium subsets for multi-parametric structural analysis1997In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 140, no 3-4, p. 305-327Article in journal (Refereed)
    Abstract [en]

    The paper discusses multi-parametric formulations for structural equilibrium problems. By seeking subsets of equilibrium states, which also fulfil extra conditions, more detailed information on structural behaviour can be obtained. These conditions can be related to analyses of, e.g. critical behaviour, imperfections or optimisation. The paper shows a general problem setting and some interesting augmentations. A solution method for the special case of one-dimensional equilibrium subsets is described. The algorithm is a generalisation of a path-following algorithm, where new methods are developed for the evaluation of the path tangent, and for the isolation of special solution points. Examples are used to show the properties and the possibilities of the ideas.

  • 260. Eriksson, A.
    Equilibrium subsets for multi-parametric structural analysis1997In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 140, no 3-4, p. 305-327Article in journal (Refereed)
    Abstract [en]

    The paper discusses multi-parametric formulations for structural equilibrium problems. By seeking subsets of equilibrium states, which also fulfil extra conditions, more detailed information on structural behaviour can be obtained. These conditions can be related to analyses of, e.g. critical behaviour, imperfections or optimisation. The paper shows a general problem setting and some interesting augmentations. A solution method for the special case of one-dimensional equilibrium subsets is described. The algorithm is a generalisation of a path-following algorithm, where new methods are developed for the evaluation of the path tangent, and for the isolation of special solution points. Examples are used to show the properties and the possibilities of the ideas.

  • 261. Eriksson, A.
    Fold lines for sensitivity analyses in structural instability1994In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 114, no 1-2, p. 77-101Article in journal (Refereed)
    Abstract [en]

    The paper describes how a two-parameter formulation of a structural equilibrium problem can be used for a more accurate description of the occurring critical states. A fold line concept is used to evaluate the dependence of these states on an added variable, describing a disturbing load case or a disturbed geometry. The concept describes the local behaviour for small disturbances, but can also be used for parameter dependence analyses, e.g. in connection with optimization algorithms. Two different augmentations of the equilibrium relations are discussed; they describe the criticality of a solution state in different ways. Numerical adoption for a general equilibrium path following algorithm is discussed. A postponed factorization method for solution of the augmented sets of equations is proposed. Two simple examples are used to show the properties and the possibilities of the fold line concept. It is concluded that the suggested numerical procedure can give a better description of critical structural behaviour, especially with respect to imperfections in the structure and idealizations in the model. © 1994.

  • 262. Eriksson, A.
    Fold lines for sensitivity analyses in structural instability1994In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 114, no 1-2, p. 77-101Article in journal (Refereed)
    Abstract [en]

    The paper describes how a two-parameter formulation of a structural equilibrium problem can be used for a more accurate description of the occurring critical states. A fold line concept is used to evaluate the dependence of these states on an added variable, describing a disturbing load case or a disturbed geometry. The concept describes the local behaviour for small disturbances, but can also be used for parameter dependence analyses, e.g. in connection with optimization algorithms. Two different augmentations of the equilibrium relations are discussed; they describe the criticality of a solution state in different ways. Numerical adoption for a general equilibrium path following algorithm is discussed. A postponed factorization method for solution of the augmented sets of equations is proposed. Two simple examples are used to show the properties and the possibilities of the fold line concept. It is concluded that the suggested numerical procedure can give a better description of critical structural behaviour, especially with respect to imperfections in the structure and idealizations in the model. © 1994.

  • 263. Eriksson, A.
    On a thin shell element for non-linear analysis, based on the isoparametric concept1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 42, no 6, p. 927-939Article in journal (Refereed)
    Abstract [en]

    This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an 'adapted' reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed. © 1992.

  • 264. Eriksson, A.
    On a thin shell element for non-linear analysis, based on the isoparametric concept1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 42, no 6, p. 927-939Article in journal (Refereed)
    Abstract [en]

    This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an 'adapted' reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed. © 1992.

  • 265. Eriksson, A.
    On accurate descriptions for primary and secondary paths in equilibrium problems1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 44, no 1-2, p. 229-242Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on ideas and expressions from the analytical elastic stability theory, have been introduced as numerical tools in a general finite element program for geometrically non-linear structural analysis. Derivatives of the tangential stiffness matrix are utilized for improved predictions in the step-wise solution of equilibrium states, for identification of critical points and for accurate descriptions of initial post-bifurcation behaviour. The methods are used in a general solution algorithm, based on a parameterizing component formulation. For some element types, analytical expressions for these derivatives can be developed. The corresponding numerical approximations, needed in other element types, are also discussed. Other practical details in the numerical implementation are given. Two numerical frame examples, showing different types of limit and bifurcation behaviours, are used to discuss the numerical properties of the methods. © 1992.

  • 266. Eriksson, A.
    On accurate descriptions for primary and secondary paths in equilibrium problems1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 44, no 1-2, p. 229-242Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on ideas and expressions from the analytical elastic stability theory, have been introduced as numerical tools in a general finite element program for geometrically non-linear structural analysis. Derivatives of the tangential stiffness matrix are utilized for improved predictions in the step-wise solution of equilibrium states, for identification of critical points and for accurate descriptions of initial post-bifurcation behaviour. The methods are used in a general solution algorithm, based on a parameterizing component formulation. For some element types, analytical expressions for these derivatives can be developed. The corresponding numerical approximations, needed in other element types, are also discussed. Other practical details in the numerical implementation are given. Two numerical frame examples, showing different types of limit and bifurcation behaviours, are used to discuss the numerical properties of the methods. © 1992.

  • 267. Eriksson, A.
    Structural instability analyses based on generalised path-following1998In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 156, no 1-4, p. 45-74Article in journal (Refereed)
    Abstract [en]

    This paper describes how quasi-static, conservative instability problems can be completely described, using generalised path-following procedures for augmented equilibrium formulations. In particular, methods for treatment of compound critical states are discussed. The numerical methods are seen as extensions to common equilibrium path methods, allowing the solution of subsets of equilibrium states, also fulfilling auxiliary relations, e.g. criticality. These formulations are in general used to describe the parameter dependence in structural response, in instability analyses and in optimisation. The paper describes the general setting of these generalised equilibrium problems, and discusses some details in their numerical treatment. Emphasis is given to the evaluation of path tangent vectors, in the presence of critical eigenvectors for the structural tangential stiffness matrix. Also, the isolation of special states, i.e. vanishing variables, turning points and exchanges of stability, is discussed. Numerical examples are used to show the possibilities and properties of the obtained solution paths, together with some aspects of the numerical procedures.

  • 268. Eriksson, A.
    et al.
    Kouhia, R.
    On step size adjustments in structural continuation problems1995In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 55, no 3, p. 495-506Article in journal (Refereed)
    Abstract [en]

    Procedures for automatic step size control in non-linear structural equilibrium analyses are discussed. Some algorithms found in the literature are briefly described and compared. The main emphasis is given to the development of criteria for detecting the need for a change of the step size as early as possible. Hence, the size of the increment is controlled during the corrector phase. Two possible procedures for this are developed and compared. Both reduce the step size in highly curved parts of an equilibrium path. The performance of the proposed algorithm is demonstrated in some numerical examples, including both geometrical and material non-linearities. The conclusions are that a procedure for reduction of the step size within the step can give improved possibilities for convergence in the iterations. The need for suitable scaling of used variables in any continuation algorithm is also emphasized. © 1995.

  • 269. Eriksson, A.
    et al.
    Kouhia, R.
    On step size adjustments in structural continuation problems1995In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 55, no 3, p. 495-506Article in journal (Refereed)
    Abstract [en]

    Procedures for automatic step size control in non-linear structural equilibrium analyses are discussed. Some algorithms found in the literature are briefly described and compared. The main emphasis is given to the development of criteria for detecting the need for a change of the step size as early as possible. Hence, the size of the increment is controlled during the corrector phase. Two possible procedures for this are developed and compared. Both reduce the step size in highly curved parts of an equilibrium path. The performance of the proposed algorithm is demonstrated in some numerical examples, including both geometrical and material non-linearities. The conclusions are that a procedure for reduction of the step size within the step can give improved possibilities for convergence in the iterations. The need for suitable scaling of used variables in any continuation algorithm is also emphasized. © 1995.

  • 270. Eriksson, A.
    et al.
    Pacoste, C.
    Symbolic software tools in the development of finite elements1999In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 72, no 4, p. 579-593Article in journal (Refereed)
    Abstract [en]

    Symbolic software has been used in a number of projects concerned with the development of finite element procedures, primarily aiming at complex, i.e. interacting and higher order instabilities, where high accuracy in formulations is required. The symbolic tools improve the efficiency and documentation of the developed procedures, in order to facilitate comparisons between different element assumptions. Beam formulations for plane and space models were developed, in total displacement and co-rotational contexts, respectively. Symbolic derivation allowed analytical verification of equivalence between certain formulations within these two contexts. Treatment of finite space rotations, based on the rotational vector makes the history-less treatment of rotations easier, which is needed in the evaluation of critical equilibrium subsets in higher-dimensional parameter space. A co-rotational viewpoint, where local element displacements can be obtained from global variables in a systematic manner, allowed different element expressions in a common framework. Different simple, linear elements have been tested with respect to computational efficiency. A field consistence approach was used to develop highly accurate beam and plane stress elements. The common element formulations, based on the matrix multiplications BTDB, is often inefficient, due to the large number of operations needed in the matrix product. Other formulations, based on an analytical integration and differentiation of the strain energy, producing explicit expressions for the stiffness terms, were considerably more efficient for certain elements.

  • 271. Eriksson, A.
    et al.
    Pacoste, C.
    Zdunek, A.
    Numerical analysis of complex instability behaviour using incremental-iterative strategies1999In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 179, no 3-4, p. 265-305Article in journal (Refereed)
    Abstract [en]

    The paper describes how quasi-static, conservative instability problems can be analysed in a multi-parametric space, using generalised path-following procedures for augmented equilibrium problems. The general formulation of such augmented equilibrium problems is discussed in some detail. The focus is set on two classes of generalised 1D paths: basic equilibrium paths and fold lines, i.e. critical subset paths. The solution methods are seen as extensions to common incremental-iterative strategies, allowing the computation of subsets of equilibrium states which also fulfil some auxiliary conditions, e.g. criticality. In this context, some emphasis is also given to the evaluation of the properties of the problem, at a certain state; the tangential stiffness is here used to evaluate - possibly multidimensional - tangent spaces, and in the isolation of special states, i.e. vanishing variables, turning points and exchanges of stability, being important aspects of instability analyses. A set of carefully chosen numerical examples demonstrate on one hand the ability of the numerical procedures to deal with complex instability phenomena, including coincident or near coincident buckling modes, modal interaction, secondary bifurcations, and, on the other hand, their versatility in performing parameter sensitivity analyses. Finally, comparisons with alternative techniques, based on asymptotic strategies, are also put forth.

  • 272.
    Eriksson, Alexander
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Effects on warm pre-stressing on cleavage fracture2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 273.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Constraint paths in non-linear structural optimization2014In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 140, p. 39-47Article in journal (Refereed)
    Abstract [en]

    Optimization of significantly non-linear structures is a demanding task. The paper discusses how boundaries of the feasible region can be followed as generalized equilibrium paths in parametric space, reflecting engineering demands on stiffness, strains and stability. Solutions on the constraint paths are then evaluated with respect to any chosen objective function. For few design parameters, this approach is efficient and robust. This is demonstrated for a pre-stressed pressurized membrane of three parameters, showing several constraint paths for the problem, and indicating how these are used in optimization. The view is often closer to engineering design analyses than the mathematical optimization settings, which often has problems in handling stability constraints.

  • 274.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Criteria for optimality in movements2006In: Journal of Biomechanics, ISSN 0021-9290, E-ISSN 1873-2380, Vol. 39, no s1, p. S54-S54Article in journal (Refereed)
  • 275. Eriksson, Anders
    Derivatives of tangential stiffness matrices for equilibrium path descriptions1991In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 32, no 5, p. 1093-1113Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on expressions from analytical elastic stability theory, are introduced as numerical tools in a general Finite Element program for geometrically non-linear structural analysis. Especially is discussed how derivatives of the tangential stiffness matrix can be utilized in several contexts in the solution algorithm. These include improved predictions for the step-wise solution of equilibrium states, identification of critical points and accurate descriptions of initial post-bifurcation behaviour. For two plane beam and bar elements, formulations have been developed giving analytical expressions for these derivatives. The corresponding numerical approximations, needed in other element types, are also discussed. The paper discusses the relative efficiency of higher order predictions in relation to these different element types and different solution strategies. Some numerical examples, showing different types of behaviour, are analysed and discussed.

  • 276.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Derivatives of tangential stiffness matrices for equilibrium path descriptions1991In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 32, no 5, p. 1093-1113Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on expressions from analytical elastic stability theory, are introduced as numerical tools in a general Finite Element program for geometrically non-linear structural analysis. Especially is discussed how derivatives of the tangential stiffness matrix can be utilized in several contexts in the solution algorithm. These include improved predictions for the step-wise solution of equilibrium states, identification of critical points and accurate descriptions of initial post-bifurcation behaviour. For two plane beam and bar elements, formulations have been developed giving analytical expressions for these derivatives. The corresponding numerical approximations, needed in other element types, are also discussed. The paper discusses the relative efficiency of higher order predictions in relation to these different element types and different solution strategies. Some numerical examples, showing different types of behaviour, are analysed and discussed.

  • 277. Eriksson, Anders
    Derivatives of tangential stiffness matrices for equilibrium path descriptions1991In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 32, no 5, p. 1093-1113Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on expressions from analytical elastic stability theory, are introduced as numerical tools in a general Finite Element program for geometrically non-linear structural analysis. Especially is discussed how derivatives of the tangential stiffness matrix can be utilized in several contexts in the solution algorithm. These include improved predictions for the step-wise solution of equilibrium states, identification of critical points and accurate descriptions of initial post-bifurcation behaviour. For two plane beam and bar elements, formulations have been developed giving analytical expressions for these derivatives. The corresponding numerical approximations, needed in other element types, are also discussed. The paper discusses the relative efficiency of higher order predictions in relation to these different element types and different solution strategies. Some numerical examples, showing different types of behaviour, are analysed and discussed.

  • 278.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Fold lines for sensitivity analyses in structural instability1994In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 114, no 1-2, p. 77-101Article in journal (Refereed)
    Abstract [en]

    The paper describes how a two-parameter formulation of a structural equilibrium problem can be used for a more accurate description of the occurring critical states. A fold line concept is used to evaluate the dependence of these states on an added variable, describing a disturbing load case or a disturbed geometry. The concept describes the local behaviour for small disturbances, but can also be used for parameter dependence analyses, e.g. in connection with optimization algorithms. Two different augmentations of the equilibrium relations are discussed; they describe the criticality of a solution state in different ways. Numerical adoption for a general equilibrium path following algorithm is discussed. A postponed factorization method for solution of the augmented sets of equations is proposed. Two simple examples are used to show the properties and the possibilities of the fold line concept. It is concluded that the suggested numerical procedure can give a better description of critical structural behaviour, especially with respect to imperfections in the structure and idealizations in the model. © 1994.

  • 279.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Multi-parametric constraint paths in optimization of pre-stressed membraneArticle in journal (Refereed)
    Abstract [en]

    Optimization of significantly non-linear structures is a demanding task. The paper discusses how boundaries of the feasible region are followed as generalized equilibrium paths in parametric space. As the optimum normally activates one or more inequalities, solutions on the constraint paths are evaluated with respect to the cost function. For few design parameters, this method is efficient and robust. The method is demonstrated for a pre-stressed pressurized membrane of three parameters, with displacement and instability constraints, and with strain limits. Numerical examples show several constraint paths for theproblem, and indicate how these are used in optimization.

  • 280.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Multi-parametric stability investigations for pressurized thin membranes2012In: ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers / [ed] J. Eberhardsteiner et al., ECCOMAS , 2012, p. 7357-7371Conference paper (Refereed)
    Abstract [en]

    The contribution describes how the quasi-static, conservative problem of pressurized space membranes, i.e., balloons, can be seen in a multi-dimensional context, and how one- and two-dimensional solution manifolds can reveal further information on the structural response. The discussed viewpoint can be seen as the natural extension of the common one-dimensional path-following methods for load-displacement traces, when additional variables are introduced to describe the parameter dependence in structural response, and in instability analyses. The paper describes the general setting of the balloon problem, and the generalized equilibrium form, with some aspects of its treatment. Numerical examples show applications of these models.

  • 281.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Multi-parametric stability investigations for pressurized thin membranes2012In: Proceedings NSCM-25 / [ed] K. Persson, J. Revstedt, G. Sandberg & M. Wallin, Lund: Lund University , 2012Conference paper (Refereed)
    Abstract [en]

    Pressurized thin membranes are frequently encountered in engineering and medical con-texts. Membranes subjected to pneumatic pressurization respond by changing volume and internal over-pressure, and can exhibit different instability conditions.  The response is significantly dependent onparameters for the loading, geometry and material. We used generalized equilibrium path and surfaceevaluation algorithms to describe and investigate the parameter dependence in different response aspects.We describe the algorithmic setting, but also give some aspects of quasi-static instability for pneumaticallypressurized thin membranes. Two examples are used to illustrate the methodology and the obtainableresults

  • 282.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Numerical modeling of thin pressurized membranes2012In: Proc CST-2012 / [ed] BHV Topping, Stirling, U.K.: Civil-Comp , 2012Chapter in book (Refereed)
    Abstract [en]

    The contribution describes how thequasi-static, conservative  problem of pressurized space membranes, i.e., balloons, can be seen in amulti-dimensional context, and how one- and two-dimensionalsolution manifolds can reveal further information on thestructural response. The discussed viewpoint can be seen as thenatural extension of the common one-dimensional path-followingmethods  for load-displacement traces, when additional variables are introduced to describe theparameter dependence in structural response, and in instabilityanalyses. The paper describes the generalsetting of the balloon problem, and the generalized equilibrium form, with some aspects of itstreatment.  Numerical examples showapplications of these models.

  • 283.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Numerical simulations of skeletal muscle mechanics2006In: Innovation in Engineering Computational Technology / [ed] Topping, BHV; Montero, G; Montenegro, R, 2006, p. 107-126Conference paper (Refereed)
    Abstract [en]

    This paper discusses the simulation of forces in a musculoskeletal system. The main focus is on the treatment of static and dynamic equilibria for a redundant force system, where the distribution of the required force for maintaining a posture or performing a movement is not unique. Reference is made to some general aspects of redundancy, and a brief discussion of the static optimisation method follows. This leads to a general framework for static optimisation, where different performance criteria can be introduced. Also dynamic equilibrium during movement is discussed, and a temporal finite element viewpoint is introduced for the treatment of target-controlled movements. This also leads to a rather general framework, where different criteria for optimal movement can be introduced. From both static and dynamic optimisation algorithms it can be concluded that existing numerical descriptions of muscular behaviour are missing some important elements, primarily related to memory aspects of the function. Finally we discuss some ideas around the types of research required in the area of muscular force production.

  • 284.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    On a thin shell element for non-linear analysis, based on the isoparametric concept1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 42, no 6, p. 927-939Article in journal (Refereed)
    Abstract [en]

    This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an 'adapted' reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed. © 1992.

  • 285.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    On accurate descriptions for primary and secondary paths in equilibrium problems1992In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 44, no 1-2, p. 229-242Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures, based on ideas and expressions from the analytical elastic stability theory, have been introduced as numerical tools in a general finite element program for geometrically non-linear structural analysis. Derivatives of the tangential stiffness matrix are utilized for improved predictions in the step-wise solution of equilibrium states, for identification of critical points and for accurate descriptions of initial post-bifurcation behaviour. The methods are used in a general solution algorithm, based on a parameterizing component formulation. For some element types, analytical expressions for these derivatives can be developed. The corresponding numerical approximations, needed in other element types, are also discussed. Other practical details in the numerical implementation are given. Two numerical frame examples, showing different types of limit and bifurcation behaviours, are used to discuss the numerical properties of the methods. © 1992.

  • 286. Eriksson, Anders
    On improved predictions for structural equilibrium path evaluations1993In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 36, no 2, p. 201-220Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures for higher-order predictions have been introduced in order to improve the convergence speed in a general finite element program for non-linear structural analysis. In addition to higher-order Taylor expansions earlier discussed, Lagrangian extrapolations and some methods commonly used for the integration of initial value problems have been introduced. The methods are used for improved predictions in the stepwise solution of equilibrium states and for accurate descriptions of the initial post-bifurcation behaviour. They are used in a general solution algorithm, based on a parameter formulation. The methods are discussed in the light of the strategies for re-creation of the tangential stiffness matrix, used for equilibrium iterations. Numerical examples, exhibiting different limit and bifurcation behaviours for trusses, frames and shells, are used to evaluate the numerical properties and efficiencies of the methods. The paper concludes that the overall efficiency in the algorithm can be improved by introduction of more accurate predictions than the standard Euler prediction. In terms of reliability combined with efficiency, an implicit generalized Simpson method is the preferred method.

  • 287.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    On improved predictions for structural equilibrium path evaluations1993In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 36, no 2, p. 201-220Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures for higher-order predictions have been introduced in order to improve the convergence speed in a general finite element program for non-linear structural analysis. In addition to higher-order Taylor expansions earlier discussed, Lagrangian extrapolations and some methods commonly used for the integration of initial value problems have been introduced. The methods are used for improved predictions in the stepwise solution of equilibrium states and for accurate descriptions of the initial post-bifurcation behaviour. They are used in a general solution algorithm, based on a parameter formulation. The methods are discussed in the light of the strategies for re-creation of the tangential stiffness matrix, used for equilibrium iterations. Numerical examples, exhibiting different limit and bifurcation behaviours for trusses, frames and shells, are used to evaluate the numerical properties and efficiencies of the methods. The paper concludes that the overall efficiency in the algorithm can be improved by introduction of more accurate predictions than the standard Euler prediction. In terms of reliability combined with efficiency, an implicit generalized Simpson method is the preferred method.

  • 288. Eriksson, Anders
    On improved predictions for structural equilibrium path evaluations1993In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 36, no 2, p. 201-220Article in journal (Refereed)
    Abstract [en]

    The paper describes how several procedures for higher-order predictions have been introduced in order to improve the convergence speed in a general finite element program for non-linear structural analysis. In addition to higher-order Taylor expansions earlier discussed, Lagrangian extrapolations and some methods commonly used for the integration of initial value problems have been introduced. The methods are used for improved predictions in the stepwise solution of equilibrium states and for accurate descriptions of the initial post-bifurcation behaviour. They are used in a general solution algorithm, based on a parameter formulation. The methods are discussed in the light of the strategies for re-creation of the tangential stiffness matrix, used for equilibrium iterations. Numerical examples, exhibiting different limit and bifurcation behaviours for trusses, frames and shells, are used to evaluate the numerical properties and efficiencies of the methods. The paper concludes that the overall efficiency in the algorithm can be improved by introduction of more accurate predictions than the standard Euler prediction. In terms of reliability combined with efficiency, an implicit generalized Simpson method is the preferred method.

  • 289. Eriksson, Anders
    On linear constraints for Newton-Raphson corrections and critical point searches in structural F.E. problems1989In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 28, no 6, p. 1317-1334Article in journal (Refereed)
    Abstract [en]

    The paper discusses the introduction of constraining equations in the tangential stiffness relation used to calculate the responses to different load cases in solution algorithms for non-linear mechanical Finite Element (F.E.) problems. An alternative to the normal two-phase solution method is discussed. This method is used to represent different iteration constraints, and in conjunction with the search for critical solution points. Numerical tests are presented, evaluating the efficiency of different iteration constraints for a model problem. Practically useful criteria for critical points are discussed. The basic methods for search of such points and some numerical aspects are discussed and evaluated for three different problems.

  • 290.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    On linear constraints for Newton-Raphson corrections and critical point searches in structural F.E. problems1989In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 28, no 6, p. 1317-1334Article in journal (Refereed)
    Abstract [en]

    The paper discusses the introduction of constraining equations in the tangential stiffness relation used to calculate the responses to different load cases in solution algorithms for non-linear mechanical Finite Element (F.E.) problems. An alternative to the normal two-phase solution method is discussed. This method is used to represent different iteration constraints, and in conjunction with the search for critical solution points. Numerical tests are presented, evaluating the efficiency of different iteration constraints for a model problem. Practically useful criteria for critical points are discussed. The basic methods for search of such points and some numerical aspects are discussed and evaluated for three different problems.

  • 291. Eriksson, Anders
    On linear constraints for Newton-Raphson corrections and critical point searches in structural F.E. problems1989In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 28, no 6, p. 1317-1334Article in journal (Refereed)
    Abstract [en]

    The paper discusses the introduction of constraining equations in the tangential stiffness relation used to calculate the responses to different load cases in solution algorithms for non-linear mechanical Finite Element (F.E.) problems. An alternative to the normal two-phase solution method is discussed. This method is used to represent different iteration constraints, and in conjunction with the search for critical solution points. Numerical tests are presented, evaluating the efficiency of different iteration constraints for a model problem. Practically useful criteria for critical points are discussed. The basic methods for search of such points and some numerical aspects are discussed and evaluated for three different problems.

  • 292. Eriksson, Anders
    ON SOME PATH-RELATED MEASURES FOR NON-LINEAR STRUCTURAL F. E. PROBLEMS.1988In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 26, no 8, p. 1791-1803Article in journal (Refereed)
    Abstract [en]

    In the paper, the solution of a non-linear structural mechanical problem is seen as a set of points along a curve in the displacement space, resulting from a continuous variation of a load parameter. The state of the structure at a specified point on the path is described by a tangent vector describing the response to a small load factor increment. For a completed, finite step, the deviation from this tangent response is described by a suggested measure. From this measure, some conclusions can be drawn concerning the iteration behavior, guiding the iteration strategy in coming steps. Two path-related stiffness measures are derived, giving information concerning the behavior of the structure. Some conclusions concerning limit load points, bifurcations, etc. can be drawn from these measures.

  • 293.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    ON SOME PATH-RELATED MEASURES FOR NON-LINEAR STRUCTURAL F. E. PROBLEMS.1988In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 26, no 8, p. 1791-1803Article in journal (Refereed)
    Abstract [en]

    In the paper, the solution of a non-linear structural mechanical problem is seen as a set of points along a curve in the displacement space, resulting from a continuous variation of a load parameter. The state of the structure at a specified point on the path is described by a tangent vector describing the response to a small load factor increment. For a completed, finite step, the deviation from this tangent response is described by a suggested measure. From this measure, some conclusions can be drawn concerning the iteration behavior, guiding the iteration strategy in coming steps. Two path-related stiffness measures are derived, giving information concerning the behavior of the structure. Some conclusions concerning limit load points, bifurcations, etc. can be drawn from these measures.

  • 294. Eriksson, Anders
    ON SOME PATH-RELATED MEASURES FOR NON-LINEAR STRUCTURAL F. E. PROBLEMS.1988In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 26, no 8, p. 1791-1803Article in journal (Refereed)
    Abstract [en]

    In the paper, the solution of a non-linear structural mechanical problem is seen as a set of points along a curve in the displacement space, resulting from a continuous variation of a load parameter. The state of the structure at a specified point on the path is described by a tangent vector describing the response to a small load factor increment. For a completed, finite step, the deviation from this tangent response is described by a suggested measure. From this measure, some conclusions can be drawn concerning the iteration behavior, guiding the iteration strategy in coming steps. Two path-related stiffness measures are derived, giving information concerning the behavior of the structure. Some conclusions concerning limit load points, bifurcations, etc. can be drawn from these measures.

  • 295.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Optimal simulations in musculoskeletal posture and movement2008In: Computer and experimental simulations in engineering and science, Vol. 1, p. 39-56Article in journal (Refereed)
  • 296.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Optimization for targeted movements2006In: Proc. III Eur. Conf. Comp. Mech.: Solids, structures and coupled problems in engineering / [ed] CA Mota-Soares, 2006Conference paper (Refereed)
  • 297.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Optimization of power input in simulated cross-country skiingManuscript (preprint) (Other academic)
    Abstract [en]

    Optimal simulations for a simplified skier in a cross-country track are considered.  The skier is modeled considering air resistance drag, friction and normal forces when following a track of cubic splines in a vertical plane. The race is modeled as a time-evolution problem, where the acting forces give the movement. Based on an assumption on how the driving  power variation along the track is limited by capacity measures, a mathematical optimization problem is formulated. This  minimizes the race time under a constraint of maximum integrated cost of the mechanical work rate. The paper discusses the mathematical and numerical formulations of the problem, and shows some aspects of discretization and accuracy. It is obvious from the simulations, that significant reductions in race time can be reached by strategically using the available power resources, rather than using a uniform work rate. With the fatigue criterion used, the conclusion from simulations is that it is advantageous  for the skier to input extra power in parts of the track where resistance is high, i.e., in up-slopes, in parts where friction is locally higher, and in parts where a head-wind is affecting the performance. Although the criterion used catches some aspects of how power production causes fatigue, it is concluded that better descriptions are needed for fatigue accumulation and reduction during a regime with different work rates.

  • 298.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Quasi-static simulations of thin membranes, aiming at stability analyses of balloon-like structures.2011In: Proc. Compdyn 2011 ECCOMAS Thematic Conference, 2011Conference paper (Refereed)
  • 299.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Structural instability analyses based on generalised path-following1998In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 156, no 1-4, p. 45-74Article in journal (Refereed)
    Abstract [en]

    This paper describes how quasi-static, conservative instability problems can be completely described, using generalised path-following procedures for augmented equilibrium formulations. In particular, methods for treatment of compound critical states are discussed. The numerical methods are seen as extensions to common equilibrium path methods, allowing the solution of subsets of equilibrium states, also fulfilling auxiliary relations, e.g. criticality. These formulations are in general used to describe the parameter dependence in structural response, in instability analyses and in optimisation. The paper describes the general setting of these generalised equilibrium problems, and discusses some details in their numerical treatment. Emphasis is given to the evaluation of path tangent vectors, in the presence of critical eigenvectors for the structural tangential stiffness matrix. Also, the isolation of special states, i.e. vanishing variables, turning points and exchanges of stability, is discussed. Numerical examples are used to show the possibilities and properties of the obtained solution paths, together with some aspects of the numerical procedures.

  • 300.
    Eriksson, Anders
    KTH, School of Engineering Sciences (SCI), Mechanics, Structural Mechanics.
    Temporal finite element descriptions in structural dynamics2004In: Proceedings of the Seventh International Conference on Computational Structures Technology / [ed] B.H.V. Topping, C.A. Mota Soares, Civil-Comp , 2004Conference paper (Refereed)
    Abstract [en]

    This paper discusses a temporal finite element approximation in the analysis of dynamics of mechanical systems, with a special emphasis on problems where a targeted control is desired. This is defined as a situation where forces are to be introduced for the movement of a structure from an initial to one or more target states, but where the behavior between these states is arbitrary. The primary applications are related to bio-mechanical simulations of skeleto-muscular systems, or to robotic analyses. By interpolating simultaneously displacements and velocities in the discrete degrees of freedom, a collocation over the time interval can be used to decide the necessary control variations. As a second step, the control can be optimized for chosen criteria on the integrated force components. By the introduced interpolation of control forces and discrete displacements, a degree of continuity is introduced in the obtained results.

    The presentation focusses on the similarity in computational formulations between several types of dynamic simulations, and sets them in a common algorithmic context. The temporal descriptions of all discrete displacement components are thereby based on a Hermitian finite element form, where each variable is represented by its value and its time differential at a set of discrete time stations. All displacement variables are thereby represented as piecewise cubic polynomials.

    Using the basic equilibrium for the stated problem, and introducing the temporal interpolation of the variables, a finite element form of the problem can be established, with elements in the time dimension, supplementing the discrete or discretized description at each time instance. A set of equations is then established by using a two-point collocation within each time element. This view allows equilibrium equations of any complexity, but is primarily suited for problems of low to moderate numbers of degrees of freedom. The acting forces consist of prescribed external forces and a priori unknown control forces. Prescribed boundary conditions add equations to the system to be solved.

    Dependent on the formulation of the problem, the solution method handles three basic classes of problems. These are distinguished by the number of boundary conditions on the displacements and velocities, and the number of free control force values. For the evolution problem, without control forces, the problem formulation must specify two values for each displacement component. For the fixed control problem, where a target state is desired, the number of free control force values is equal to the number of excessive boundary conditions, and their values can be determined. For an optimal control problem, the number of free control force values is higher than the number of excessive displacement conditions, allowing the optimization of their values.

    For all the three classes of problems, a set of equation is established. In the optimal control problem, the set will add equations of optimality, increasing the size of the problem; a general algorithm can, however, be easily established, where only the number of prescribed displacement values and the number of free control force components decide the used method.

    Performed tests indicate that the developed viewpoint and algorithm can be efficient in the study of complex, but primarily small to moderate size problems, with an improved continuity in the description of motion, and a good stability in dynamic solution. Comparing accuracy and computational effort, the method is efficient for a small problem, compared to Euler and Newmark methods, [1], and comparable to a Runge-Kutta 4th order method.

    The method avoids the common shooting procedure to find a target displacement state, by solving for all discrete time stations at once. For a target controlled non-linear mechanism problem, the convergence with discretization is studied, and shown to be quick, given that a reasonably good initial approximation can be introduced. The method also allows full Newton iterations, leading to high accuracy in results. Alternative local optima in control force cost are discussed, starting from a well-known problem, [2].

    With a sparse matrix for the established system, the efficiency of the method can in many cases be improved. Ongoing work expands the method to allow redundant force systems, limits in control force values, and interpolation of activation measures in the muscular system, rather than in forces themselves.

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