Finite element method
The finite element method, is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, electromagnetic potential; the analytical solution of these problems require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations; the method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements; the simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. Studying or analyzing a phenomenon with FEM is referred to as finite element analysis; the subdivision of a whole domain into simpler parts has several advantages: Accurate representation of complex geometry Inclusion of dissimilar material properties Easy representation of the total solution Capture of local effects.
A typical work out of the method involves dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, can be calculated from the initial values of the original problem to obtain a numerical answer. In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are partial differential equations. To explain the approximation in this process, FEM is introduced as a special case of Galerkin method; the process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE.
The residual is the error caused by the trial functions, the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with a set of algebraic equations for steady state problems, a set of ordinary differential equations for transient problems; these equation sets are the element equations. They are linear if the underlying PDE is linear, vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method. In step above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes; this spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system.
The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its practical application, known as finite element analysis. FEA as applied in engineering is a computational tool for performing engineering analysis, it includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system. FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations.
For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear. Another example would be in numerical weather prediction, where it is more important to have accurate predictions over developing nonlinear phenomena rather than calm areas. While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering, its development can be traced back to the work by R. Courant in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is connected with name of Leonard Oganesyan. In China, in the 1950s and early 1960s, based on the computations of dam constructions, K. Feng proposed a systematic numerical method for solving partial differential equations; the method was called the finite difference method based on variation principle, another independent invention of the finite element met
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function; this article considers linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative exist. Assume that there is a map A from a function space F 1 to another function space F 2 and a function f ∈ F 2 so that f is the image of u ∈ F 1 i.e. F = A. A differential operator is represented as a linear combination, finitely generated by u and its derivatives containing higher degree such as P = ∑ | α | ≤ m a α D α, where the set of non-negative integers, α =, is called a multi-index, | α | = α 1 + α 2 + ⋯ + α n called length, a α are functions on some open domain in n-dimensional space and D α = D 1 α 1 D 2 α 2 ⋯ D n α n; the derivative above is one as functions or, distributions or hyperfunctions and D j = − i ∂ ∂ x j or sometimes, D j = ∂ ∂ x j.
The most common differential operator is the action of taking derivative. Common notations for taking the first derivative with respect to a variable x include: d d x, D, D x, ∂ x; when taking higher, nth order derivatives, the operator may be written: d n d x n, D n, D x n, or ∂ x n. The derivative of a function f of an argument x is sometimes given as either of the following: ′ f ′; the D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form ∑ k = 0 n c k D k in his study of differential equations. One of the most seen differential operators is the Laplacian operator, defined by Δ = ∇ 2 = ∑ k = 1 n ∂ 2 ∂ x k 2. Another differential operator is the Θ operator, or theta operator, defined by Θ = z d d z; this is sometimes called the homogeneity operator, because its eigenfunctions are the monomials in z: Θ = k z k, k = 0, 1, 2, … In n variables the homogeneity operator is given by Θ = ∑ k = 1 n x k ∂ ∂ x k. As in one variable
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two; because such relations are common, differential equations play a prominent role in many disciplines including engineering, physics and biology. In pure mathematics, differential equations are studied from several different perspectives concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas. If a closed-form expression for the solution is not available, the solution may be numerically approximated using computers; the theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. Differential equations first came into existence with the invention of calculus by Newton and Leibniz.
In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations: d y d x = f d y d x = f x 1 ∂ y ∂ x 1 + x 2 ∂ y ∂ x 2 = y He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695; this is an ordinary differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. The problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur, in which he based his reasoning on Newton's law of cooling, that the flow of heat between two adjacent molecules is proportional to the small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat; this partial differential equation is now taught to every student of mathematical physics. For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance.
The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, air resistance may be modeled as proportional to the ball's velocity; this means that the ball's acceleration, a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation and verifying its validity. Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, Homogeneous/Inhomogeneous; this list is far from exhaustive. An ordinary differential equation is an equation containing an unknown function of one real or complex variable x, its derivatives, some
Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.
For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.
Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.
He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.
Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old
Social science is a category of academic disciplines, concerned with society and the relationships among individuals within a society. Social science as a whole has many branches; these social sciences include, but are not limited to: anthropology, communication studies, history, human geography, linguistics, political science, public health, sociology. The term is sometimes used to refer to the field of sociology, the original "science of society", established in the 19th century. For a more detailed list of sub-disciplines within the social sciences see: Outline of social science. Positivist social scientists use methods resembling those of the natural sciences as tools for understanding society, so define science in its stricter modern sense. Interpretivist social scientists, by contrast, may use social critique or symbolic interpretation rather than constructing empirically falsifiable theories, thus treat science in its broader sense. In modern academic practice, researchers are eclectic, using multiple methodologies.
The term "social research" has acquired a degree of autonomy as practitioners from various disciplines share in its aims and methods. The history of the social sciences begins in the Age of Enlightenment after 1650, which saw a revolution within natural philosophy, changing the basic framework by which individuals understood what was "scientific". Social sciences came forth from the moral philosophy of the time and were influenced by the Age of Revolutions, such as the Industrial Revolution and the French Revolution; the social sciences developed from the sciences, or the systematic knowledge-bases or prescriptive practices, relating to the social improvement of a group of interacting entities. The beginnings of the social sciences in the 18th century are reflected in the grand encyclopedia of Diderot, with articles from Jean-Jacques Rousseau and other pioneers; the growth of the social sciences is reflected in other specialized encyclopedias. The modern period saw "social science" first used as a distinct conceptual field.
Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative. Auguste Comte used the term "science sociale" to describe the field, taken from the ideas of Charles Fourier. Following this period, there were five paths of development that sprang forth in the social sciences, influenced by Comte on other fields. One route, taken was the rise of social research. Large statistical surveys were undertaken in various parts of the United States and Europe. Another route undertaken was initiated by Émile Durkheim, studying "social facts", Vilfredo Pareto, opening metatheoretical ideas and individual theories. A third means developed, arising from the methodological dichotomy present, in which social phenomena were identified with and understood; the fourth route taken, based in economics, was developed and furthered economic knowledge as a hard science. The last path was the correlation of knowledge and social values. In this route and prescription were non-overlapping formal discussions of a subject.
Around the start of the 20th century, Enlightenment philosophy was challenged in various quarters. After the use of classical theories since the end of the scientific revolution, various fields substituted mathematics studies for experimental studies and examining equations to build a theoretical structure; the development of social science subfields became quantitative in methodology. The interdisciplinary and cross-disciplinary nature of scientific inquiry into human behaviour and environmental factors affecting it, made many of the natural sciences interested in some aspects of social science methodology. Examples of boundary blurring include emerging disciplines like social research of medicine, neuropsychology and the history and sociology of science. Quantitative research and qualitative methods are being integrated in the study of human action and its implications and consequences. In the first half of the 20th century, statistics became a free-standing discipline of applied mathematics.
Statistical methods were used confidently. In the contemporary period, Karl Popper and Talcott Parsons influenced the furtherance of the social sciences. Researchers continue to search for a unified consensus on what methodology might have the power and refinement to connect a proposed "grand theory" with the various midrange theories that, with considerable success, continue to provide usable frameworks for massive, growing data banks; the social sciences will for the foreseeable future be composed of different zones in the research of, sometime distinct in approach toward, the field. The term "social science" may refer either to the specific sciences of society established by thinkers such as Comte, Durkheim and Weber, or more to all disciplines outside of "noble science" and arts. By the late 19th century, the academic social sciences were constituted of five fields: jurisprudence and amendment of the law, health and trade, art. Around the start of the 21st century, the expanding domain of economics in the social sciences has been described as economic imperialism.
The social science disciplines are branches of knowledge taught and researched at the college or university level. Social science disciplines are defined and rec
Natural science is a branch of science concerned with the description and understanding of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two main branches: physical science. Physical science is subdivided into branches, including physics, chemistry and earth science; these branches of natural science may be further divided into more specialized branches. In Western society's analytic tradition, the empirical sciences and natural sciences use tools from formal sciences, such as mathematics and logic, converting information about nature into measurements which can be explained as clear statements of the "laws of nature"; the social sciences use such methods, but rely more on qualitative research, so that they are sometimes called "soft science", whereas natural sciences, insofar as they emphasize quantifiable data produced and confirmed through the scientific method, are sometimes called "hard science".
Modern natural science succeeded more classical approaches to natural philosophy traced to ancient Greece. Galileo, Descartes and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives and presuppositions overlooked, remain necessary in natural science. Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, minerals, so on. Today, "natural history" suggests observational descriptions aimed at popular audiences. Philosophers of science have suggested a number of criteria, including Karl Popper's controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity and quality control, such as peer review and repeatability of findings, are amongst the most respected criteria in the present-day global scientific community; this field encompasses a set of disciplines.
The scale of study can range from sub-component biophysics up to complex ecologies. Biology is concerned with the characteristics and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment; the biological fields of botany and medicine date back to early periods of civilization, while microbiology was introduced in the 17th century with the invention of the microscope. However, it was not until the 19th century. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole; some key developments in biology were the discovery of genetics. Modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the fundamental chemistry of life, while cellular biology is the examination of the cell. At a higher level and physiology look at the internal structures, their functions, of an organism, while ecology looks at how various organisms interrelate.
Constituting the scientific study of matter at the atomic and molecular scale, chemistry deals with collections of atoms, such as gases, molecules and metals. The composition, statistical properties and reactions of these materials are studied. Chemistry involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications. Most chemical processes can be studied directly in a laboratory, using a series of techniques for manipulating materials, as well as an understanding of the underlying processes. Chemistry is called "the central science" because of its role in connecting the other natural sciences. Early experiments in chemistry had their roots in the system of Alchemy, a set of beliefs combining mysticism with physical experiments; the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, Antoine Lavoisier, who developed the theory of the Conservation of mass. The discovery of the chemical elements and atomic theory began to systematize this science, researchers developed a fundamental understanding of states of matter, chemical bonds and chemical reactions.
The success of this science led to a complementary chemical industry that now plays a significant role in the world economy. Physics embodies the study of the fundamental constituents of the universe, the forces and interactions they exert on one another, the results produced by these interactions. In general, physics is regarded as the fundamental science, because all other natural sciences use and obey the principles and laws set down by the field. Physics relies on mathematics as the logical framework for formulation and quantification of principles; the study of the principles of the universe has a long history and derives from direct observation and experimentation. The formulation of theories about the governing laws of the universe has been central to the study of physics from early on, with philosophy yielding to systematic, quantitative experimental testing and observation as the source of verification. Key historical developments in physics include Isaac Newton's theory of universal g