Variational Views in Mechanics

This book provides a comprehensive survey of interactions between the calculus of variations and theoretical and applied mechanics, covering topics such as topology optimization, identification of material properties, optimal control, plastic flows, gradient polyconvexity, obstacle problems, quasi-monotonicity, and variational views in mechanics. It is a valuable reference for researchers in mathematics, solid-states physics, and mechanical, civil, and materials engineering.

Format: Paperback / softback
Length: 309 pages
Publication date: 09 February 2023
Publisher: Springer Nature Switzerland AG

This comprehensive volume offers a timely and insightful exploration of the intricate interplay between the calculus of variations and theoretical and applied mechanics. Since its initial publication in a special issue of the Journal of Optimization Theory and Applications (184(1),2020) titled "Calculus of Variations in Mechanics and Related Fields," the chapters have undergone extensive revisions to provide a comprehensive and up-to-date overview of the field. The diverse range of topics covered in this volume offers researchers a comprehensive perspective on the problems in mechanics that can be approached using variational techniques. This makes it an invaluable resource for scholars engaged in research in the realm of mechanics. Furthermore, the book presents exciting ideas for potential future research directions, demonstrating the immense potential of mastering these fundamental mathematical tools for a wide array of exciting applications.

The specific topics addressed within this volume include:

Topology Optimization: This chapter delves into the optimization of geometric structures, aiming to find the optimal shape or configuration that minimizes a given objective function. It employs techniques from the calculus of variations to derive the necessary conditions for optimality and solve the associated optimization problems.

Identification of Material Properties: This chapter focuses on developing efficient methods for identifying the material properties of complex structures, such as composites or heterogeneous materials. It explores the use of variational techniques, including finite element analysis and optimization algorithms, to accurately model and predict the behavior of these materials under various loading conditions.

Optimal Control: This chapter explores the theory and application of optimal control, a branch of mathematics that deals with the design of control systems for achieving desired performance objectives. It discusses the use of variational techniques to find the optimal control laws that minimize the cost or maximize the performance of mechanical systems.

Plastic Flows: This chapter examines the mathematical models and computational techniques used to study the flow of plastic materials, such as polymers or metals. It discusses the use of variational methods to describe the deformation and flow of these materials and to optimize their properties for specific applications.

Gradient Polyconvexity: This chapter explores the concept of gradient polyconvexity, a mathematical framework that describes the behavior of functions defined on convex sets. It discusses the use of variational techniques to analyze and optimize problems involving gradient-based optimization algorithms.

Obstacle Problems: This chapter focuses on the mathematical modeling and analysis of obstacle problems, which involve the movement of objects in constrained environments. It discusses the use of variational techniques to find optimal trajectories or configurations that minimize the energy or cost associated with the obstacle avoidance.

Quasi-Monotonicity: This chapter examines the concept of quasi-monotonicity, a property that ensures the non-decreasing or non-increasing behavior of a function or system. It discusses the use of variational techniques to analyze and optimize problems involving quasi-monotonicity constraints.

Variational Views in Mechanics: This chapter provides an overview of the variational approach to mechanics, which views mechanical systems as energy minimizers. It discusses the fundamental principles of variational mechanics and its applications to various mechanical problems, including solid mechanics, fluid mechanics, and thermodynamics.

This volume will appeal to researchers in mathematics, solid-states physics, and mechanical, civil, and materials engineering. It offers a comprehensive and interdisciplinary perspective on the applications of the calculus of variations in mechanics and provides valuable insights into the latest developments and research trends in this field.

Weight: 504g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030900533
Edition number: 1st ed. 2021