anti periodic solutions to a class of nonlinear differential equations in hilbert space abstract a broad class of nonlinear non monotone anti periodic boundary value problems in a hilbert space is considered as a preliminary step the existence uniqueness and continuous dependence upon data of anti periodic solutions to some first and second order evolution equations associated to odd noncoercive monotone operators is established applications of the theory to nonlinear partial differential equations are also discussed

a finite element method for stability problems in finite elasticity in this paper a finite element method is developed to treat stability problems in finite elasticity for this purpose the constitutive equations are formulated in principal stretches which allows a general representation of the derivatives of the strain energy function with respect to the principal stretches these results can then be used to derive an efficient numerical scheme for the computation of singular points

on a class of second order anti periodic boundary value problems of concern is the existence and uniqueness of anti periodic solutions to a class of abstract nonlinear second order differential equations the investigation relies on the theory of m accretive operators in banach or hilbert spaces examples of boundary value problems for ordinary and partial differential equations are also discussed

a recursive description of pro p galois groups abstract in this note we study a modified version of the elementary type conjecture for pro p galois groups to be precise let g p f be the galois group of the maximal galois p extension of a field f containing a primitive p th root of unity under some natural assumptions concerning valuation rings of f and also orderings of f when p 2 we prove that g p f can be obtained from suitable closed subgroups using a finite number of free pro p products and semidirect products

computational two mode asymptotic bifurcation theory combined with hyper dual numbers and applied to plate shell buckling abstract for error free computation of higher order derivatives of a complex mathematical expression composed of elementary functions hyper dual numbers hdns are receiving close attention in computational engineering and sciences differently from conventional finite differences hdns numerically evaluate for instance the exact first order and second order derivatives of the stiffness matrix with respect to nodal degrees of freedom dof in nonlinear finite element fe analysis one of the most promising applications of hdns is the asymptotic structural stability theory which usually requires in conventional series expansions an unrealistically large number of second order derivative of the stiffness matrix to predict the instability behavior of large scale fe models this computational bottleneck may be avoided along with so called classical lyapunov schmidt koiter condensation by a novel idea proposed in the present study the proposed two mode computational asymptotic theory is implementable in engineering practice to algebraically identify snap through and path branching in stability problems one more associated innovation is a graphical solution of two simultaneous equations resulting from the asymptotic expansions this sophisticated solution idea utilizing a popular mathematical tool matlab is widely applicable to quickly visualize the location and number of existing solutions more specifically the present paper formulates a two mode asymptotic bifurcation theory in combination with hdns and presents an examination of the proposed computational theory on stability problems of plates and shells near a precisely computed singular point on the equilibrium path the incremental displacement field is represented in two modes and one mode is the critical eigenvector corresponding to the zero eigenvalue of the singular tangent stiffness matrix the other one is a linear combination of all other non critical eigenvectors and respectively constitute the homogeneous and particular solutions of the linearized equilibrium equations established in the singular point two perturbation parameters and work as weights of and respectively to find possible equilibria around the singular point the nonlinear equilibrium equations are expanded into asymptotic series of and to use hdns to compute 20 coefficients including first order and second order derivatives of the stiffness matrix in polynomials the simultaneous polynomial equations of and are solved by using the matlab graphic operations the obtained results in numerical examples well predict the stability behavior in excellent agreement with solutions available in conventional fe stability analysis the two mode asymptotic bifurcation theory combined with hdns is sufficient to diagnose snap through asymmetric branching unstable and stable symmetric bifurcation in structural stability problems

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

a highly accurate 1st and 2nd order differentiation scheme for hyperelastic material models based on hyper dual numbers abstract in this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models which are derived in terms of the first and second derivatives of a strain energy function this numerical scheme provides a compact model independent framework which means that once the framework is implemented any other hyperelastic material model can be incorporated by solely modifying the energy function the method is based on the numerical calculation of strain energy derivatives using hyper dual numbers and thus referred to as hyper dual step derivative hdsd the hdsd does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values furthermore it enables the calculation of derivatives of arbitrary order this is a great advantage compared to other numerical approaches as e g the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values another alternative the complex step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values but it only provides first derivatives and is thus not able to calculate stresses and moduli at once in this paper representative numerical examples using an anisotropic model are provided showing the performance of the proposed method in detail an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme furthermore examples demonstrate the robustness and simple implementation of the hdsd scheme in finite element software it turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time

anti periodic solutions for semilinear evolution equations in this paper we use the homotopy method to establish the existence and uniqueness of anti periodic solutions for the nonlinear anti periodic problem a t x bx f t a e t r x t t x t where a t x is a nonlinear map and b is a bounded linear operator from r n to r n sufficient conditions for the existence of the solution set are presented also we consider the nonlinear evolution problems with a perturbation term which is multivalued we show that for this problem the solution set is nonempty and weakly compact in w 1 2 i r n for the case of convex valued perturbation and prove the existence theorems of anti periodic solutions for the nonconvex case all illustrative examples are provided

hill top branching its asymptotically expanded and visually solved bifurcation equations abstract hill top branching htb is an exceptional stability problem in which the bifurcation point bp coincides exactly with a limit point lp in such cases snap through and branching instability occur simultaneously then htb is often obscured in snap through behavior and is often overlooked in stability design the compound instability is in fact not well realized in the community of computational mechanics when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined no report of the relevant literature has established diagnosis of path branching or proposed a branch switching procedure furthermore few htb models are available in the literature this report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue there might be a single or two critical eigenvectors of the singular stiffness matrix higher derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper dual numbers hdns the resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as matlab from the locations of existing real roots displayed on a graphic monitor the crossover situation of path branching is visually clear above and below the hill top for the present study three htb examples are computed to verify the proposed algorithm strategy using hdns and graphical tools the obtained numerical results show excellent agreement with analytical and finite element predictions the proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric htb in structural stability problems and function well in computational engineering problems

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

modified stiffness iteration to pinpoint multiple bifurcation points abstract a modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented the iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones two different transformation matrices are employed to modify the stiffness matrix one is a non orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix the other is an orthogonal one which is chosen with reference to the symmetry of the system under consideration the use of the non orthogonal transformation matrix is a key idea termed stiffness amplification method in this paper whereas the orthogonal matrix is based on block diagonalization which is becoming popular in group theoretic analysis of symmetric structures extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points

robust numerical calculation of tangent moduli at finite strains based on complex step derivative approximation and its application to localization analysis abstract an extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard fem software this method is based on the complex step derivative approximation csda approach the csda is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e g finite difference approximation therefore the csda can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision implementation details of the robust numerical approximation of tangent moduli from stress calculations using the csda are given and their performance is illustrated through representative examples involving finite deformations in addition to that we focus on the determination of material instabilities therefore an accompanying localization analysis is performed where the acoustic tensor is directly computed from the approximation of the moduli it is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation on the other hand the csda approach provides high accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field