Shigeru Nakagiri

Finite element interval analysis of external loads identified by displacement input with uncertainty

A finite element formulation is proposed for the interval estimation of the nodal forces identified by displacement input with uncertainty. The stiffness matrix of the structural system in problem of linear, elastic body under static loading is assumed to be known and determinate. This assumption enables us to derive the governing equation of the nodal forces to be identified with respect to the nominal displacement input from the stiffness equation based on the Moore-Penrose generalized inverse solution of the unknown remnant displacements. We assume that the uncertain errors involved in the nodal displacements obtained somehow by measurement and used as input are confined in a convex hull. The effect of the uncertainty on the identified nodal forces is evaluated by the sensitivity analysis of the forces with respect to the uncertain displacements. How to estimate the scatter of the identified nodal forces corresponding to the uncertain displacements confined in the convex hull is devised by means of the Lagrange multiplier method. The validity of the proposed method for identification and interval estimation of the nodal forces is shown by the numerical example of an elastic square plate in the plane stress state and subject to in-plane loading.

Worst case estimation of homology design by convex analysis

Abstract The methodology of homology design is investigated for optimum design of advanced structures, for which the achievement of delicate tasks by the aid of active control system is demanded. The proposed formulation of homology design, based on the finite element sensitivity analysis, necessarily requires the specification of external loadings. The formulation to evaluate the worst case for homology design caused by uncertain fluctuation of loadings is presented by means of the convex model of uncertainty, in which uncertainty variables are assigned to discretized nodal forces and are confined within a conceivable convex hull given as a hyperellipse. The worst case of the distortion from objective homologous deformation is estimated by the Lagrange multiplier method searching the point to maximize the error index on the boundary of the convex hull. The validity of the proposed method is demonstrated in a numerical example using the eleven-bar truss structure.

Biomechanical studies on shape effect of hydroxyapatite artificial root upon surrounding jawbone

To study the shape effect upon surrounding tissue of artificial dental root made of synthetic hydroxyapatite, this paper deals with the numerical analysis of the artificial root on functioning jawbone with the aid of the finite element method (FEM). The stress distribution around artificial roots in the shape of a cylinder, a cone, and three types of corrugated cone, including the newly tailored type implanted in the mandibular molar region, was analyzed in the plane strain state. The numerical results showed that the stress distribution was sensitive to the artificial root shape, and that the stress state was distributed in mitigatory way around the roots of the newly tailored form. The pattern of osteogenesis in the animal experiment and the finite element analysis (FEA) pattern showed a close correlation. Osteogenesis was assumed to occur in the weak or moderate stress distribution zone. The principal stress trajectory pattern in the lamina dura around the tailored artificial root was indicated as being either parallel or normal to the root surface. From this study, the biomechanical property of the tooth can be identified as a vehicle of mastication forces which disperse stresses moderately and equally upon surrounding tissues. Also, the periodontal ligament can be identified as a converting system of principal stress trajectories.

Biomechanical studies on newly tailored artificial dental root.

Artificial roots must carry multiple forces during mastication. Stress distribution around a root depends upon the shape, material, and function of the root. Therefore, for biomechanical studies on artificial roots, triad research on the material, shape, and functional effect upon surrounding tissue is essential. For dental implants, there are two different functional systems against the masticatory force, i.e., gomphosis and ankylosis on osseointegration. Stress analyses of functioning new type (gomphosis) artificial roots were carried out in mandibular and maxilla models to study the triad effect using finite element analysis. The authors have already reported histological and biomechanical studies on the shape and functional effect. To observe the material effect biomechanically, artificial roots made of sintered hydroxyapatite and zirconium oxide were analyzed in the models. Thereafter, animal experiments using dogs were carried out to observe bone formation around artificial roots made of hydroxyapatite and zirconium oxide in the mandible and maxilla. The following results were obtained: The patterns of stress distribution around artificial roots of two different materials were not too different, and were exclusively dependent upon the root shape and structure of the jawbone. Around the artificial roots, bone formation coincided with a moderate stress distributing zone and principal stress trajectories. Through these experiments, the following conclusions were obtained: (a) Osteogenesis around artificial roots coincides with the stress distribution patterns. (b) Stress distribution patterns are dependent very little upon material properties but upon both the artificial root shape and the structure of the jawbone. (c) Optimization of the artificial root shape can be obtained by FEA in the models.

Biomechanical Research on Junction System of Bone with Biomaterials

Regarding the junction of bioceramics with original bone, which have quite different material constants of Youngs modulus and Poissons ratio from each other, synostosis (ankylosis) cannot be obtained under severe loading conditions. Therefore, it is necessary to introduce a new junction system for the interface between the biomaterial mechanical organ and original bone. The jointing system of dental root to jawbone reflects on the function against mastication. The interface between different mechanical organs with different materials necessitates a specific juncture system under severe loading because of the disparity of material constants. The authors already reported the result of studies on the shape effect of artificial roots in functioning jawbone by means of finite element analysis. Studies on the functional effect of artificial roots in undulated shape were carried out biomechanically by means of finite element analysis using models to investigate an effective juncture system between bone and biomaterials. The results of finite element analysis were compared with the findings obtained from histological specimens. To observe the juncture state of bioceramics with tubular bone cortex, tubular apatite artificial bone was implanted in the femur of a dog. From these studies, the following results and conclusions were obtained: (a) The fibrous juncture system around bioceramics has an important role, after which the principal stress trajectories are converted; and (b) optimal undulated morphology compatible to the artificial bones juncture system by means of fibrous ligament is essential for remodeling of the bone around the artificial skeletal bone.

Stochastic Analysis of a Beam on Random Foundation with Uncertain Damping Subjected to a Moving Load

The steady state vibration of the rail on the foundation with uncertain stiffness and damping is investigated. The stochastic finite element method is employed to the analysis. Basic formulation is reviewed first. Relevant matrices for the analysis are newly formulated. The variances of deflection and bending moment of the rail caused by the uncertainties are evaluated by the first-order perturbation technique and first-order second-moment method. The validity of the method is demonstrated by numerical examples.

Design change of frame structure to enhance structural reliability

Abstract A method of design change aimed at the enhancement of structural reliability is formulated. The perturbation-based stochastic finite element method is employed to evaluate the reliability index of the advanced first-order second-moment method. The enhancement of the reliability index is devised by means of the minimal design change from a baseline design, while the shift of the reliability index is incorporated in terms of an equality constraint condition. The validity of the proposed method is verified by the numerical example of a two-storeyed portal frame with uncertain Youngs modulus, for which the moment of inertia of the frame members is taken as design variables.

Structural Synthesis for the Worst Case Mitigation Based on the Convex Set of Uncertainty.

A non-probabilistic methodology is presented for the design to enhance structural integrity against the uncertainty in structural parameters. The uncertainty is represented by a convex set of uncertainty variables, in which the fluctuation of uncertainty variables is bounded within a convex hull given as a hyperellipse without consideration of the probability distribution inside. The worst case is determined in terms of unfavorable index defined so as to indicate the degree of harmfulness for the structural integrity. Use is made of the finite element sensitivity analysis and the Lagrange multiplier method for the search of the nucertainty variables to raise the worst case. The worst unfavorable index thus identified is mitigated less than the tolerance limit in line with the structural synthesis based on the first-order approximation of the worst unfavorable index with respect to design variables. The validity of the proposed formulation is investigated through the numerical example concerning with a three-storied portal frame with three braces.

Identification of Distributed Young's Modulus and Interval Analysis Due to Uncertain Strain Input.

A formulation is proposed for identification of spatially distributed Youngs modulus based on strain input. The strain and its sensitivity with respect to isotropic Youngs modulus are analysed by the finite element method with the initial guess of Youngs modulus in order to obtain the firstorder approximation of strain change. The numerical strain, obtained by the finite element analysis, is compared with such strain input as measured strain, and the current guess of Youngs modulus is renewed iteratively so as to eliminate the deviation between the numerical strain and strain input until the Youngs modulus is settled by the renewal converged. The fluctuation of the identified Youngs modulus that arises from uncertain error involved in the strain input is estimated in the form of interval by means of the convex modeling of the error and Lagrange multiplier method. The validity of the present formulation is shown by the numerical example of axial distribution of Youngs modulus identified by the skin strain of beam bending.

Passive Vibration Shape Control of Frame Structure by Homology Design.

A formulation of homology design is presented for passive control of a structural shape under dynamic excitations. Prescribed geometrical property in a part of structure is kept both in vibrant state and standstill one by homology design for the purpose to reduce control cost expected for high quality of performance. Time-history analysis is carried out by the Newmark β method to evaluate the error index, which indicates the time-average of the difference from the objective homologous vibration shape. The error index is minimized by iterative renewals using the finite element sensitivity analysis and the Newtons method. The validity of the proposed formulation is demonstrated in a numerical example concerning with planar lattice frame, a square portion of which is kept horizontal even in a vibrant state by homology design.