Generation Of Stiffness Matrix Research Articles

Machine-learning and digital-twins for rapid evaluation and design of injected vaccine immune-system responses

A computational framework is developed that researchers in the field can easily implement and subsequently use as an efficient tool to study the immune-response to a vaccine injection. There are three main components to this work: Part I-Digital-twin construction: An approach is developed that efficiently simulates the time-transient proliferation of cells/antibodies (proteins) and regulator/antigens (deactivated toxin) to an injected vaccine within tissue possessing complex heterogeneous microstructure. Here, we use the terms “cells” and “antibodies”, as well as “regulator” and “antigen” interchangeably. The approach utilizes two strongly-coupled conservation laws: (a) Conservation Law 1: comprises (a) rate of change of cells/antibodies, (b) cellular/antibody migration, (c) cellular/antibody proliferation controlled by a cell/antibody mitosis regulating chemical (antigen), (d) cell/antibody apoptosis and (b) Conservation Law 2: comprises (a) rate of change of the cell/antibody mitosis chemical regulator/antigen, (b) regulator/antigen diffusion, (c) regulator production by cells/antibody and (d) regulator/antigen decay. Part II-Efficient computation: A technique based on a voxel (3D “volume pixels”) representation of tissue microstructures and corresponding digital solution methods is developed for the calculations, which avoids computationally expensive steps involved in usual Finite Element procedures such as topologically conforming meshing, mapping, volume integration, stiffness matrix generation and matrix-based solution methods. The process proceeds by converting the tissue microstructure into voxels. The problem then becomes “digital” on a regular “voxel-grid”, directly manipulating voxel values, allowing extremely fast methods to be used to construct derivatives and to iteratively solve the system with minimal memory requirements. Part III-Machine-learning: The rapid and efficient computation allows for many vaccines to be tested quickly and uses a genomic-based Machine-Learning Algorithm to optimize the system. This is particularly useful for rapid design of next-generation vaccines and boosters for disease strain mutations. Numerical examples are provided to illustrate the results, with the overall goal being to provide a computational framework to rapidly design and deploy a vaccine for a targeted response.

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

SummaryWe describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

Rapid Voxel-Based Digital-Computation for Complex Microstructured Media

A technique based on voxel (3D “volume pixels”) representation of microstructures and corresponding digital solution methods is developed for representative volume element type calculations, which avoids computationally expensive steps involved in usual Finite Element procedures such as topologically conforming meshing, mapping, volume integration, stiffness matrix generation and matrix-based solution methods. The process proceeds by converting the material microstructure into voxels. The problem then becomes “digital” on a regular “voxel-grid”, directly manipulating voxel values, allowing extremely fast methods to be used to construct derivatives and the solve the system. In this paper, we illustrate the process of voxelization, derivative construction and solution, and also supply an analysis of the operation counts. Numerical examples are provided to illustrate the approach.

Time efficiency and error estimation in generating element stiffness matrices of plane triangular elements using Universal Matrix Method and Gauss-Quadrature

This paper is aimed at generating element stiffness matrices for the family of triangular elements using Universal Matrix Method and performing a CPU time efficiency analysis and comparison of stiffness coefficients deviation (error). The interpolation functions of the field variable function (displacement) are to be integrated explicitly once and for all when geometric transformation function is linear, to give the constant universal matrices. This makes the generation of stiffness matrix a simple addition of universal matrices rather than integration of functions.

Geometriškai netiesinių rėminių konstrukcijų 2D elemento tangentinės standumo matricos sudarymo ypatumai

Geometriškai netiesinių konstrukcijų baigtinių elementų metodo tangentinių matricų komponenčių analitinės išraiškos nėra išsamiai pateiktos literatūroje. Dažniausiai pateikiamos tik mažų poslinkių standumo matricos. Tad sprendžiant įvairius konstrukcijų analizės bei projektavimo uždavinius ir norint kuo tiksliau aprašyti konstrukcijos tikrąjį įtemptąjį deformuotą būvį, būtina turėti visiškai aprašytą tangentinės matricos analitinę išraišką. Šiame straipsnyje nagrinėjama geometriškai netiesinio 2D rėminio elemento tangentinės standumo matricos sudarymo metodika naudojantis diskretizuoto kūno pilnutinės potencinės energijos stacionarumo sąlygomis vertinant tik tarpelemetines sąveikos jėgas. Gautos vidinių jėgų vektoriaus-funkcijos išvestinė pagal mazginius poslinkius ir yra tangentinė standumo matrica. Pateiktos rėminės konstrukcijos 2D elemento tangentinę standumo matricą sudarančių matricų analitinės išraiškos, išreikštos mazginiais poslinkiais. Siūloma metodika buvo patikrinta atliekant simbolinius skaičiavimus MatLAB skaičiavimo komplekso terpėje. Gauta analitinė tangentinės standumo matricos išraiška.

Fixed grid finite elements in elasticity problems

A fixed grid representation of the finite element domain is used to solve elasticity problems. An integration between CAD and FEM systems is made by using this fixed grid representation. Some considerations about FG approximation and stiffness matrix generation are presented. The element stiffness matrix for an element in the mesh is obtained as a factor of a standard element stiffness matrix. A least squares local approximation of stress field is used to calculate the stress at boundary elements. A problem with an analytical solution is used to measure the displacement and stress error. Several meshes and several geometrical configurations of the problem are used to test the reliability of the fixed grid finite element method.

Formulation and implementation of a singular anisotropic finite element

The formulation and implementation of a singular finite element for analyzing homogeneous anistropic materials is presented in this paper. Lekhnitskii's stress function method is used to formulate the boundary value problem with the stress function expressed as a Laurent series. The development of the element stiffness matrix and the method of integrating the element to conventional displacement based finite element programs is shown. The stiffness matrix generation is based on a least squates collocation technique to satisfy displacement continuity boundary conditions at the element interface. Implementation of the element is demonstrated for cracked anisotropic materials subjected to inplane loading. Center cracked, on and off-axis coupons under tensile loading are analyzed using the element. It is shown that the stress distributions and intensity factors compare well with those obtained using other methods.

Experience with the conjugate gradient method for stress analysis on a data parallel supercomputer

AbstractThe storage requirements and performance consequences of a few different data parallel implementations of the finite element method for domains discretized by three‐dimensional brick elements are reviewed. Letting a processor represent a nodal point per unassembled finite element yields a concurrency that may be one to two orders of magnitude higher for common elements than if a processor represents an unassembled finite element. The former representation also allows for higher order elements with a limited amount of storage per processor. A totally parallel stiffness matrix generation algorithm is presented. The equilibrium equations are solved by a conjugate gradient method with diagonal scaling. The results from several simulations designed to show the dependence of the number of iterations to convergence upon the Poisson ratio, the finite element discretization and the element order are reported. The domain was discretized by three‐dimensional Lagrange elements in all cases. The number of iterations to convergence increases with the Poisson ratio. Increasing the number of elements in one special dimension increases the number of iterations to convergence, linearly. Increasing the element order p in one spatial dimension increases the number of iterations to convergence as pα, where α is 1·4–1·5 for the model problems.

Thermal stress analysis of reentry vehicle nosetips at angle of attack

The ASAAS (Asymmetric Stress Analysis of Axisymmetric Solids) computer program is applied to the prediction of thermal stresses in a reentry vehicle nosetip subjected to asymmetric temperature distributions and angle-of-attack loadings during reentry. This three-dimensional stress analysis computer program has the unique capability of properly accounting for the circumferential variation of temperature dependent material properties. It is based on a meridional finite element discretization of an axisymmetric solid combined with a Fourier series representation of circumferentially dependent variables. In contrast with similar methods where material properties cannot be expressed as a function of temperature, this method requires that the large system of equations be solved simultaneously. This is handled efficiently in ASAAS by several newly developed approaches to stiffness matrix generation and simultaneous equation solution. The nosetip analyzed is fabricated from an orthotropic, temperature dependent graphite material that is susceptible to thermal shock. Analyses are performed at several points in a trajectory and the effects of both aerodynamic heating and pressure loading are considered. The complete states of temperature, stress, strain and displacement throughout the graphite material are obtained as a product of ASAAS. Results are presented in the form of meridional contour plots at selected circumferential stations. The usefulness of ASAAS in performing angle-of-attack analysis of nosetips is evaluated by comparison to an axisymmetric analysis based on single ray heating. In addition, the convergence of solutions with increasing numbers of harmonics is demonstrated and computer run times are discussed.