It is difficult to clarify the activity of swallowing-related muscles and the movement of organs. To represent the relationship between muscle activity and organ movement, a numerical simulator that reflects muscle contraction is necessary. However, no continuum organ models can identify and reflect muscle activity. In this study, a novel method was developed for obtaining the time change in the activities of all muscles involved in hyoid excursion. An organ model was developed based on four-dimensional computed tomography (4DCT) images acquired using a 320-column area detector CT, and the organ movement was analysed using a particle method. To identify the muscle activity, an evaluation function was defined and calculated considering the errors of velocities and positions of the hyoid bone, and the muscle activities to minimise the evaluation function were calculated using gradient descent. The results showed that the hyoid movement represented by the simulation model and the objective movement obtained from 4DCT images were in good agreement. The identified muscle activities were discussed through a comparison with the literature on electromyography measurements and the results of simulations using other methods. The proposed continuum organ model with muscles was considered able to clarify both muscle activity and organ movement.

A method for finding the optimum distribution of the slit density in a shoe upper, such that the contact pressure with the foot approaches an ideal distribution is presented in this paper. In a shoe upper made of warp knitting, slits are made by discontinuously knitting the weft under a uniform arrangement of the warp fiber. In this study, the shoe upper is modeled as an orthotropic hyperelastic body in which the slit density is defined as the design variable for controlling the intermediate density for the maximum and minimum slit densities. Referring to a formulation of a topology optimization problem of the density variation type, an inverse problem for finding the optimum distribution of the slit density is formulated with the objective and constraint cost functions defined by the squared error norm between the actual and ideal contact pressures and the deviation of the rate of the material with the maximum slit density from a limit value, respectively. This problem is solved in the same way as a topology optimization problem. The validity of this method is confirmed by numerical examples using a simple finite element model, which resembles the domain around the big toe.

A class of optimization problems which are formulated using solutions to boundary value problems of partial differential equations (PDEs) as equality constraints is called PDE-constrained optimization problems. Specifically, when the variable in the optimization problem is given by a function defined in the PDE, the problems are classified as PDE-constrained non-parametric optimization problems. The topology and shape optimization problems of density and domain variation type, respectively, are included in this category. This paper is written to convey the idea of the author who worked on this topic for over 30 year, unlike the usual review papers. After showing real images of the problems, the author’s theoretical image is introduced. Based on the theoretical image, basic questions about the structure of the problems when viewed as a function optimization problem are proposed. The author’s answers to these basic questions are presented in the next section. Based on the understanding, some numerical results in which anxious phenomena are observed are presented and their causes are discussed. The expectation of applying the basic theory to real-world problems is presented using examples. Applying the theory of shape optimization problem of domain variation type, a problem finding the optimum shape of a sole in sports shoes that maximizes the stability under keeping the ideal cushioning is introduced. Applying the theory of topology optimization problem of density variation type, a problem identifying the muscle activity in a tongue during swallowing is explained. From these examples, it is concluded that PDE-constrained non-parametric optimization problems can be effectively applied to real-world.

In this study, a formulation of the inverse problem to identify the shape of a hyperelastic body in which the vibration mode is similar to the swimming mode of a fish is presented. This research aims to demonstrate the possibility of creating a fish robot that swims with a vibration mode in the water when excited with a vibration generator embedded in the body. Before this study, Chancharoen et al. attempted to formulate an inverse problem as a shape optimization problem for a linear elastic body without considering water. In this study, a cost function was defined by the squared error norm of the vibration mode and the ideal swimming mode. The result of a numerical example confirmed that the approach decreased the cost function, but the obtained vibration model was different from the ideal one. In contrast, in this study, a hyperelastic body is used to approach the actual movement of a fish. Using this replacement, the finite deformation theory is employed to formulate the periodic vibration of the hyperelastic body. The cost function is formulated using the squared error norm of the finite deformation in the cycle. Its shape derivative is evaluated using the solutions of the periodic vibration problem and its adjoint problem with respect to the cost function. To solve the shape optimization problem, an iterative scheme based on the H1 gradient method for domain variation problems is used. The effectiveness of this approach is illustrated through numerical examples using finite-element models.

This paper presents a formulation to identify muscle activities from the variation in shapes of organs during swallowing. We assume that each organ consists of a three-dimensional hyperelastic body, and the contraction movement of the muscle is caused by a contractive inelastic stress in the organ. A function distributed in the organ domain to control the magnitude of the inelastic stress is chosen as a design variable in the same manner as the density in the topology optimization problem of density variation type. The identification problem is formulated as a problem of determining the design variable that minimizes an objective cost function defined by the squared L2 norm of the reaction force in the normal direction on the boundary when an enforced displacement to fit the varied boundary of the organ and the inelastic stress modeling the muscle activity are applied. The finite deformation problem of the hyperelastic body is analyzed using the finite element method. The direction of the muscle fiber is assumed to be the direction of the minimum principal stress obtained as the solution to the finite deformation problem. The solution to the identification problem is presented based on a scheme using the H1 gradient method for the topology optimization problem of density variation type. A numerical example using a previously developed model of the tongue is introduced to demonstrate the effectiveness of the proposed approach.

This study presents methods to develop a three-dimensional numerical foot model and to identify the loading condition that is used to design a stable sole for running shoes. In a previous study, the authors proposed a method to optimize the shape of the sole to increase stability while maintaining the cushioning property. In the problem formulation, the loading condition was given as a boundary force distributed on the top surface of the sole. The aim of this study is to replace the loading condition with the force and moment at the origin of the ankle joint coordinate (AJC) system by modeling a foot with a finite element model. A finite element model of a foot is constructed using X-ray CT image data, and consists of bony structures, soft tissue, and plantar fascia. The plantar fascia is set at the bottom of the bony structures. The force and moment used in the finite element analysis are identified by inverse dynamic analysis using an experimental measurement in the practical operation of the ground reaction force (GRF) when the GRF in the direction of the foot length becomes minimum. In the finite element analysis, the finite deformation containing the contact condition between the bottom surface of the foot and the ground representing a sole made of resin is considered. For an index of the shoe stability, we define a heel eversion angle (HEA) by the rotational angle of the heel with respect to an axis in the foot length direction and evaluate it by finite element analysis. The validity of the finite element foot model as well as the force and moment obtained in this study are confirmed based on the agreement in the HEA results between the experiment and finite element analysis.

This paper presents a method to reduce the computational time required to solve shape optimization problems. A volume minimization problem under the mean compliance constraint is chosen as an example of the shape optimization problem. To solve this problem, an iterative algorithm based on the $$H^1$$ gradient method is considered as a conventional approach. In this study, we attempt to use a method of model order reduction for solving the linear elasticity problem based on the idea by Karhunen-Loève expansion (KLE). We consider the displacements obtained by the conventional method to be sampling data of a random variable; the orthonormal bases of KLE are defined as eigenfunctions of the eigenvalue problem obtained as the optimality condition of the variance maximization problem for the random variable. The feasibility of the proposed method is illustrated by testing the numerical scheme to a linear elastic body of the connecting rod type.

This paper proposes a solution to a multi-material robust topology optimization problem of density type considering material uncertainties based on H1 gradient method. A material interpolation with respect to the density is introduced using the rational approximation of material properties (RAMP) and generalized it for the case with an arbitrary number of materials. Material uncertainty is considered by introducing random variables in the material interpolation scheme. The probability density functions of the random variables are assumed to be given. The topology optimization is formulated using the density which is given by a sigmoid function of the design variable. A weighted sum of the mean and standard deviation of the mean compliance is used as the objective function to control the tradeoff between optimality and robustness. To evaluate statistical moments of the objective function effectively, the univariate dimension reduction (UDR) and the Gauss-type quadrature sampling are introduced. A scheme to solve the robust topology optimization problem is presented using an iterative algorithm based on the H1 gradient method for reshaping. Examples of a two-dimensional cantilever beam under various material uncertainty exhibit the efficiency and flexibility of the approach. The accuracy of UDR is validated by comparing the results to the Monte Carlo approach.

In Chap. 8 we looked at problems for obtaining the optimal topologies of continua with the densities of continua set to be the design variable. In this chapter, we shall look at the type of shape optimization problems in which the boundary of a continuum varies. The key theory of numerical solution shown in this chapter is published in the paper (Azegami, Trans. Jpn. Soc. Industrial Appl. Math. 23(2), 83–138 (2014)). In this book, we shall look at the theory used there by comparing it to the contents shown in Chaps. 1 to 7.

This paper presents a formulation to identify the muscle activities from shape variation of organs in a swallow motion. Using a tongue model consisting of hyper elastic body and multi-muscle fibers, we consider that the shape variation is realized by the compulsory displacement on the boundary, and that muscle activities are assumed as initial stresses generated in the directions of the muscle fibers. The identification problem of the muscle activities is formulated using the magnitudes of the initial stresses as the design variables and the L2 squared norm of the reaction force on the boundary by the compulsory displacement as the cost function of minimization. Solution of the problem is presented based on the scheme using the H1 gradient method for topology optimization problem of density type. A numerical example using a previously developed data is introduced to show the effectiveness of the present approach.

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the L^2 -distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.

We have seen in Chaps. 5 and 6 how boundary problems of partial differential equations are constructed and how they are solved. If we compare them to the optimum design problems looked at in Chap. 1, they correspond to state determination problems. From this chapter, we finally start thinking about optimum design problems targeting the shape or topology of a domain in which the boundary value problem is defined. In this chapter, we construct an abstract problem common to both problems, and explore the ways to solve them.

In Chap. 2, we discussed the conditions satisfied by a local minimum point (the required conditions of a local minimum point) and the conditions which guarantee it to be a minimum point (sufficient conditions for a minimum point) under a finite-dimensional vector space setting. No detailed explanation, however, was provided regarding the method (solution) for finding the local minimum point. In this chapter, we would like to address this ensuing matter. The computational formulation associated to such a problem is called an optimization problem or a mathematical programming problem, and active research is being conducted in the academic field referred to as operations research (OR). Here, we will consider algorithms while showing results that are theoretically obtained or ways to deal with the solution of optimization problems. Much of the content covered here is also valid for abstract optimal design problems in Chap. 7. In fact, in Chap. 7 we will see how the same algorithms can be adapted for function spaces.

As seen in Chap. 1, optimal design problems are optimization problems whose state equations are considered as equality constraints. In Chap. 1, we have considered design variables and state variables as elements of a finite-dimensional vector space. However, in this book, our main interest focuses on the shape optimization problem of continuum. In this case, boundary value problems of partial differential equations, such as linear elastic bodies and Stokes flow field, are included in the equality constraints as state equations.

The main topic of this book is optimal design. In order to understand the mathematical structures involved in our study, we will begin by examining two simple problems. Upon finishing this book, the reader should be able to understand that even shape optimization problems of continuum structures possess the same formulation as the problems dealt with in this chapter. Moreover, even if the target continuum is changed from linear elastic body or flow field dealt in this book, the reader will recognize that their corresponding shape optimization problems maintain the fundamental structures introduced in this book.

In Chap. 5, covering several boundary value problems of elliptic partial differential equations, we saw that the existence of their unique solutions could be guaranteed using solutions of the weak form. These will be referred to as the exact solutions. Exact solutions can be found analytically if the shape of the domain is somewhat simple such as a rectangle or an ellipse. However, difficulties arise for domains whose shape may have arbitrarily moved as examined in this book. In order to solve a shape optimization problem, even if an exact solution is not possible, one can resort to a numerical analysis method to obtain an approximate solution.

From this chapter, we finally examine shape optimization problems in continua. Firstly, let us think about a problem seeking the appropriate arrangement of holes in a domain where the boundary value problem of a partial differential equation is defined. Such a problem is known as the topology optimization problem. Here, the term topology refers to the study of geometrical properties and spatial relation of objects unaffected by the continuous change of their shape or size. In mathematics, two mathematical objects are said to belong to the same topology if they are images of two homotopic maps; that is, if one can be continuously deformed into the other. Therefore, letting n be a natural number, a set of n-connected domains are regarded as belonging to the same homotopy groups. Here, the term “topology optimization” in the topology optimization problem refers to the determination of the connectivity of the design domain that optimizes an object’s material distribution through insertion and arrangement of holes in its structure. However, as will be explained in detail later, the shape of the holes actually becomes the target. Therefore, the problems dealt with in this chapter also become included in shape optimization problems in a wider sense. In this book, it will be referred to as the topology optimization problem in the sense that topology is also in the scope of the design.

Chapter 1 investigated explicit optimal design problems and illustrated different approaches for obtaining optimality conditions. Terminology and results utilized in optimization theory were also used. This chapter presents a systematic discussion of optimization theory.

In last chapters, we looked at the theory of solutions relating to optimization problems in finite-dimension. From this chapter onward, we shall consider optimization problems where the design variables are of function of time and space.

The present paper describes a shape optimization procedure for designing running shoes, focusing on two mechanical properties, namely, the shock absorption and the stability keeping the right posture. These properties are evaluated from two deformations of a sole at characteristic timings during running motion. We define approximate planes for the deformations of sole’s upper boundary by least squares method. Using the planes, we choose the tilt angle in the shoe width direction at the mid stance phase of running motion as an objective function representing the stability, and the sunk amount at the contact phase of running motion as a constraint function representing the shock absorption. We assume that the sole is a bonded structure of soft and hard hyper-elastic materials, and the bonding and side boundaries are variable. In this study, we apply the formulation of nonparametric shape optimization to the sole considering finite deformation and contact condition of the bottom of the sole with the ground. Shape derivatives of the cost (objective and constraint) functions are obtained using the adjoint method. The H 1 gradient method using these shape derivatives is applied as an iterative algorithm. To solve this optimization problem, we developed a computer program combined with some commercial softwares. The validity of the optimization method is confirmed by numerical examples.